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Focus (geometry) From Wikipedia, the free encyclopedia Contents

1 Circle 1 1.1 Terminology ...... 1 1.2 History ...... 2 1.3 Analytic results ...... 2 1.3.1 Length of circumference ...... 2 1.3.2 Area enclosed ...... 2 1.3.3 Equations ...... 4 1.3.4 Tangent lines ...... 8 1.4 Properties ...... 9 1.4.1 Chord ...... 9 1.4.2 Sagitta ...... 10 1.4.3 Tangent ...... 10 1.4.4 Theorems ...... 11 1.4.5 Inscribed angles ...... 12 1.5 Circle of Apollonius ...... 12 1.5.1 Cross-ratios ...... 13 1.5.2 Generalised circles ...... 13 1.6 Circles inscribed in or circumscribed about other ﬁgures ...... 14 1.7 Circle as limiting case of other ﬁgures ...... 14 1.8 Squaring the circle ...... 14 1.9 See also ...... 14 1.10 References ...... 14 1.11 Further reading ...... 15 1.12 External links ...... 15

2 Conic section 16 2.1 History ...... 17 2.1.1 Menaechmus and early works ...... 17 2.1.2 Apollonius of Perga ...... 17 2.1.3 Al-Kuhi ...... 17 2.1.4 Omar Khayyám ...... 17 2.1.5 Europe ...... 17 2.2 Features ...... 18

i ii CONTENTS

2.3 Construction ...... 18 2.4 Properties ...... 18 2.4.1 Intersection at infinity ...... 18 2.4.2 Degenerate cases ...... 19 2.4.3 Eccentricity, focus and directrix ...... 19 2.4.4 Generalizations ...... 19 2.4.5 In other areas of mathematics ...... 20 2.5 Cartesian coordinates ...... 21 2.5.1 Discriminant classification ...... 21 2.5.2 Matrix notation ...... 21 2.5.3 As slice of quadratic form ...... 22 2.5.4 Eccentricity in terms of parameters of the quadratic form ...... 22 2.5.5 Standard form ...... 22 2.5.6 Invariants of conics ...... 23 2.5.7 Modified form ...... 23 2.6 Homogeneous coordinates ...... 24 2.7 Polar coordinates ...... 24 2.8 Pencil of conics ...... 25 2.9 Intersecting two conics ...... 25 2.10 Applications ...... 26 2.11 See also ...... 26 2.12 Notes ...... 26 2.13 References ...... 27 2.14 External links ...... 27

3 Focus (geometry) 39 3.1 Conic sections ...... 40 3.1.1 Defining conics in terms of two foci ...... 40 3.1.2 Defining conics in terms of a focus and a directrix ...... 40 3.1.3 Defining conics in terms of a focus and a directrix circle ...... 40 3.1.4 Astronomical significance ...... 40 3.2 Cartesian and Cassini ovals ...... 41 3.3 Generalization ...... 41 3.4 Confocal curves ...... 41 3.5 References ...... 41

4 Locus (mathematics) 43 4.1 Commonly studied loci ...... 43 4.2 Proof of a locus ...... 44 4.3 Examples ...... 44 4.3.1 First example ...... 44 4.3.2 Second example ...... 45 CONTENTS iii

4.3.3 Third example ...... 46 4.3.4 Fourth example ...... 46 4.4 See also ...... 46 4.5 References ...... 46

5 Point (geometry) 48 5.1 Points in Euclidean geometry ...... 48 5.2 Dimension of a point ...... 48 5.2.1 Vector space dimension ...... 49 5.2.2 Topological dimension ...... 49 5.2.3 Hausdorﬀ dimension ...... 50 5.3 Geometry without points ...... 50 5.4 Point masses and the Dirac delta function ...... 50 5.5 See also ...... 50 5.6 References ...... 51 5.7 External links ...... 51 5.8 Text and image sources, contributors, and licenses ...... 52 5.8.1 Text ...... 52 5.8.2 Images ...... 54 5.8.3 Content license ...... 55 Chapter 1

Circle

This article is about the shape and mathematical concept. For other uses, see Circle (disambiguation).

A circle is a simple shape in Euclidean geometry. It is the set of all points in a plane that are at a given distance from a given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from a given point is constant. The distance between any of the points and the centre is called the radius. A circle is a simple closed curve which divides the plane into two regions: an interior and an exterior. In everyday use, the term “circle” may be used interchangeably to refer to either the boundary of the figure, or to the whole figure including its interior; in strict technical usage, the circle is the former and the latter is called a disk. A circle may also be defined as a special ellipse in which the two foci are coincident and the eccentricity is 0, or the two-dimensional shape enclosing the most area per unit perimeter squared, using calculus of variations.

A circle is a plane ﬁgure bounded by one line, and such that all right lines drawn from a certain point within it to the bounding line, are equal. The bounding line is called its circumference and the point, its centre. — Euclid. Elements Book I. [1]

1.1 Terminology

• Arc: any connected part of the circle.

• Centre: the point equidistant from the points on the circle.

• Chord: a line segment whose endpoints lie on the circle.

• Circumference: the length of one circuit along the circle, or the distance around the circle.

• Diameter: a line segment whose endpoints lie on the circle and which passes through the centre; or the length of such a line segment, which is the largest distance between any two points on the circle. It is a special case of a chord, namely the longest chord, and it is twice the radius.

• Passant: a coplanar straight line that does not touch the circle.

• Radius: a line segment joining the centre of the circle to any point on the circle itself; or the length of such a segment, which is half a diameter.

• Sector: a region bounded by two radii and an arc lying between the radii.

• Segment: a region, not containing the centre, bounded by a chord and an arc lying between the chord’s endpoints.

• Secant: an extended chord, a coplanar straight line cutting the circle at two points.

1 2 CHAPTER 1. CIRCLE

• Semicircle: an arc that extends from one of a diameter’s endpoints to the other. In non-technical common usage it may mean the diameter, arc, and its interior, a two dimensional region, that is technically called a half-disk. A half-disk is a special case of a segment, namely the largest one.

• Tangent: a coplanar straight line that touches the circle at a single point.

1.2 History

The word “circle” derives from the Greek κίρκος/κύκλος (kirkos/kuklos), itself a metathesis of the Homeric Greek κρίκος (krikos), meaning “hoop” or “ring”.[2] The origins of the words "circus" and "circuit" are closely related. The circle has been known since before the beginning of recorded history. Natural circles would have been observed, such as the Moon, Sun, and a short plant stalk blowing in the wind on sand, which forms a circle shape in the sand. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy, and calculus. Early science, particularly geometry and astrology and astronomy, was connected to the divine for most medieval scholars, and many believed that there was something intrinsically “divine” or “perfect” that could be found in circles.[3][4] Some highlights in the history of the circle are:

• 1700 BCE – The Rhind papyrus gives a method to ﬁnd the area of a circular ﬁeld. The result corresponds to 256 [5] ⁄81 (3.16049...) as an approximate value of π.

• 300 BCE – Book 3 of Euclid’s Elements deals with the properties of circles.

• In Plato's Seventh Letter there is a detailed definition and explanation of the circle. Plato explains the perfect circle, and how it is different from any drawing, words, definition or explanation.

• 1880 CE– Lindemann proves that π is transcendental, eﬀectively settling the millennia-old problem of squaring the circle.[6]

1.3 Analytic results

1.3.1 Length of circumference

Further information: Circumference

The ratio of a circle’s circumference to its diameter is π (pi), an irrational constant approximately equal to 3.141592654. Thus the length of the circumference C is related to the radius r and diameter d by:

C = 2πr = πd.

1.3.2 Area enclosed

Main article: Area of a disk

As proved by Archimedes, the area enclosed by a circle is equal to that of a triangle whose base has the length of the circle’s circumference and whose height equals the circle’s radius,[7] which comes to π multiplied by the radius squared: 1.3. ANALYTIC RESULTS 3

The compass in this 13th-century manuscript is a symbol of God’s act of Creation. Notice also the circular shape of the halo

Area = πr2.

Equivalently, denoting diameter by d, 4 CHAPTER 1. CIRCLE

Circular piece of silk with Mongol images

πd2 Area = ≈ 0.7854d2, 4 that is, approximately 79 percent of the circumscribing square (whose side is of length d). The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality.

1.3.3 Equations

Cartesian coordinates

In an x–y Cartesian coordinate system, the circle with centre coordinates (a, b) and radius r is the set of all points (x, y) such that

(x − a)2 + (y − b)2 = r2. 1.3. ANALYTIC RESULTS 5

Circles in an old Arabic astronomical drawing.

This equation, known as the Equation of the Circle, follows from the Pythagorean theorem applied to any point on the circle: as shown in the diagram to the right, the radius is the hypotenuse of a right-angled triangle whose other sides are of length |x − a| and |y − b|. If the circle is centred at the origin (0, 0), then the equation simpliﬁes to x2 + y2 = r2.

The equation can be written in parametric form using the trigonometric functions sine and cosine as x = a + r cos t, y = b + r sin t where t is a parametric variable in the range 0 to 2π, interpreted geometrically as the angle that the ray from (a, b) to (x, y) makes with the positive x-axis. An alternative parametrisation of the circle is:

2t x = a + r . 1 + t2 6 CHAPTER 1. CIRCLE

Tughrul Tower from inside

1 − t2 y = b + r 1 + t2 In this parametrisation, the ratio of t to r can be interpreted geometrically as the stereographic projection of the line passing through the centre parallel to the x-axis (see Tangent half-angle substitution). However, this parametrisation works only if t is made to range not only through all reals but also to a point at inﬁnity; otherwise, the bottom-most point of the circle would be omitted. In homogeneous coordinates each conic section with the equation of a circle has the form x2 + y2 − 2axz − 2byz + cz2 = 0.

It can be proven that a conic section is a circle exactly when it contains (when extended to the complex projective plane) the points I(1: i: 0) and J(1: −i: 0). These points are called the circular points at inﬁnity.

Polar coordinates

In polar coordinates the equation of a circle is:

2 − − 2 2 r 2rr0 cos(θ ϕ) + r0 = a where a is the radius of the circle, (r, θ) is the polar coordinate of a generic point on the circle, and (r0, ϕ) is the polar coordinate of the centre of the circle (i.e., r0 is the distance from the origin to the centre of the circle, and φ is the anticlockwise angle from the positive x-axis to the line connecting the origin to the centre of the circle). For a circle centred at the origin, i.e. r0 = 0, this reduces to simply r = a. When r0 = a, or when the origin lies on the circle, the equation becomes 1.3. ANALYTIC RESULTS 7

Area = r2

Circle Area = π×r 2

Area enclosed by a circle = π × area of the shaded square

r = 2a cos(θ − ϕ).

In the general case, the equation can be solved for r, giving

√ − 2 − 2 2 − r = r0 cos(θ ϕ) a r0 sin (θ ϕ),

Note that without the ± sign, the equation would in some cases describe only half a circle.

Complex plane

In the complex plane, a circle with a centre at c and radius (r) has the equation |z − c| = r . In parametric form this can be written z = reit + c . The slightly generalised equation pzz+gz+gz = q for real p, q and complex g is sometimes called a generalised circle. This becomes the above equation for a circle with p = 1, g = −c, q = r2 −|c|2 , since |z −c|2 = zz −cz −cz +cc . Not all generalised circles are actually circles: a generalised circle is either a (true) circle or a line. 8 CHAPTER 1. CIRCLE

0.5

(x,y)

0.5 1 1.5 2 r

-0.5 (a,b)

-1

-1.5

Circle of radius r = 1, centre (a, b) = (1.2, −0.5)

1.3.4 Tangent lines

Main article: Tangent lines to circles

The tangent line through a point P on the circle is perpendicular to the diameter passing through P. If P = (x1, y1) and the circle has centre (a, b) and radius r, then the tangent line is perpendicular to the line from (a, b) to (x1, y1), so it has the form (x1 − a)x + (y1 – b)y = c. Evaluating at (x1, y1) determines the value of c and the result is that the equation of the tangent is

(x1 − a)x + (y1 − b)y = (x1 − a)x1 + (y1 − b)y1 or

2 (x1 − a)(x − a) + (y1 − b)(y − b) = r .

If y1 ≠ b then the slope of this line is dy x − a = − 1 . dx y1 − b 1.4. PROPERTIES 9

This can also be found using implicit diﬀerentiation. When the centre of the circle is at the origin then the equation of the tangent line becomes

2 x1x + y1y = r , and its slope is dy x = − 1 . dx y1

1.4 Properties

• The circle is the shape with the largest area for a given length of perimeter. (See Isoperimetric inequality.)

• The circle is a highly symmetric shape: every line through the centre forms a line of reﬂection symmetry and it has rotational symmetry around the centre for every angle. Its symmetry group is the orthogonal group O(2,R). The group of rotations alone is the circle group T.

• All circles are similar.

• A circle’s circumference and radius are proportional. • The area enclosed and the square of its radius are proportional. • The constants of proportionality are 2π and π, respectively.

• The circle which is centred at the origin with radius 1 is called the unit circle.

• Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.

• Through any three points, not all on the same line, there lies a unique circle. In Cartesian coordinates, it is possible to give explicit formulae for the coordinates of the centre of the circle and the radius in terms of the coordinates of the three given points. See circumcircle.

1.4.1 Chord

• Chords are equidistant from the centre of a circle if and only if they are equal in length.

• The perpendicular bisector of a chord passes through the centre of a circle; equivalent statements stemming from the uniqueness of the perpendicular bisector are:

• A perpendicular line from the centre of a circle bisects the chord. • The line segment through the centre bisecting a chord is perpendicular to the chord.

• If a central angle and an inscribed angle of a circle are subtended by the same chord and on the same side of the chord, then the central angle is twice the inscribed angle.

• If two angles are inscribed on the same chord and on the same side of the chord, then they are equal.

• If two angles are inscribed on the same chord and on opposite sides of the chord, then they are supplementary.

• For a cyclic quadrilateral, the exterior angle is equal to the interior opposite angle.

• An inscribed angle subtended by a diameter is a right angle (see Thales’ theorem).

• The diameter is the longest chord of the circle.

• If the intersection of any two chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then ab = cd. 10 CHAPTER 1. CIRCLE

• If the intersection of any two perpendicular chords divides one chord into lengths a and b and divides the other chord into lengths c and d, then a2 + b2 + c2 + d2 equals the square of the diameter.[8] • The sum of the squared lengths of any two chords intersecting at right angles at a given point is the same as that of any other two perpendicular chords intersecting at the same point, and is given by 8r 2 – 4p 2 (where r is the circle’s radius and p is the distance from the center point to the point of intersection).[9] • The distance from a point on the circle to a given chord times the diameter of the circle equals the product of the distances from the point to the ends of the chord.[10]:p.71

1.4.2 Sagitta

The sagitta is the vertical segment.

• The sagitta (also known as the versine) is a line segment drawn perpendicular to a chord, between the midpoint of that chord and the arc of the circle. • Given the length y of a chord, and the length x of the sagitta, the Pythagorean theorem can be used to calculate the radius of the unique circle which will ﬁt around the two lines:

y2 x r = + . 8x 2

Another proof of this result which relies only on two chord properties given above is as follows. Given a chord of length y and with sagitta of length x, since the sagitta intersects the midpoint of the chord, we know it is part of a diameter of the circle. Since the diameter is twice the radius, the “missing” part of the diameter is (2r − x) in length. Using the fact that one part of one chord times the other part is equal to the same product taken along a chord intersecting the first chord, we find that (2r − x)x = (y / 2)2. Solving for r, we find the required result.

1.4.3 Tangent

• A line drawn perpendicular to a radius through the end point of the radius lying on the circle is a tangent to the circle. • A line drawn perpendicular to a tangent through the point of contact with a circle passes through the centre of the circle. • Two tangents can always be drawn to a circle from any point outside the circle, and these tangents are equal in length. • If a tangent at A and a tangent at B intersect at the exterior point P, then denoting the centre as O, the angles ∠BOA and ∠BPA are supplementary.

1 • If AD is tangent to the circle at A and if AQ is a chord of the circle, then ∠DAQ = ⁄2arc(AQ). 1.4. PROPERTIES 11

1.4.4 Theorems

B C A

Secant-secant theorem

See also: Generalised circle

If C is the midpoint of the segment AB, then the collection of points P satisfying the Apollonius condition

|AP | |AC| |BP | = |BC| 14 CHAPTER 1. CIRCLE is not a circle, but rather a line. Thus, if A, B, and C are given distinct points in the plane, then the locus of points P satisfying the above equation is called a “generalised circle.” It may either be a true circle or a line. In this sense a line is a generalised circle of inﬁnite radius.

1.6 Circles inscribed in or circumscribed about other ﬁgures

In every triangle a unique circle, called the incircle, can be inscribed such that it is tangent to each of the three sides of the triangle.[14] About every triangle a unique circle, called the circumcircle, can be circumscribed such that it goes through each of the triangle’s three vertices.[15] A tangential polygon, such as a tangential quadrilateral, is any convex polygon within which a circle can be inscribed that is tangent to each side of the polygon.[16] A cyclic polygon is any convex polygon about which a circle can be circumscribed, passing through each vertex. A well-studied example is the cyclic quadrilateral. A hypocycloid is a curve that is inscribed in a given circle by tracing a ﬁxed point on a smaller circle that rolls within and tangent to the given circle.

1.7 Circle as limiting case of other ﬁgures

The circle can be viewed as a limiting case of each of various other ﬁgures:

• A Cartesian oval is a set of points such that a weighted sum of the distances from any of its points to two ﬁxed points (foci) is a constant. An ellipse is the case in which the weights are equal. A circle is an ellipse with an eccentricity of zero, meaning that the two foci coincide with each other as the centre of the circle. A circle is also a diﬀerent special case of a Cartesian oval in which one of the weights is zero.

• x n y n A superellipse has an equation of the form a + b = 1 for positive a, b, and n. A supercircle has b = a. A circle is the special case of a supercircle in which n = 2.

• A Cassini oval is a set of points such that the product of the distances from any of its points to two ﬁxed points is a constant. When the two ﬁxed points coincide, a circle results.

• A curve of constant width is a figure whose width, defined as the perpendicular distance between two distinct parallel lines each intersecting its boundary in a single point, is the same regardless of the direction of those two parallel lines. The circle is the simplest example of this type of figure.

1.8 Squaring the circle

Squaring the circle is the problem, proposed by ancient geometers, of constructing a square with the same area as a given circle by using only a ﬁnite number of steps with compass and straightedge. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental number, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coeﬃcients.

1.9 See also

1.10 References

[1] OL7227282M 1.11. FURTHER READING 15

[2] krikos, Henry George Liddell, Robert Scott, A Greek-English Lexicon, on Perseus

[3] Arthur Koestler, The Sleepwalkers: A History of Man’s Changing Vision of the Universe (1959)

[4] Proclus, The Six Books of Proclus, the Platonic Successor, on the Theology of Plato Tr. Thomas Taylor (1816) Vol.2, Ch.2, “Of Plato”

[5] Chronology for 30000 BC to 500 BC. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.

[6] Squaring the circle. History.mcs.st-andrews.ac.uk. Retrieved on 2012-05-03.

[7] Measurement of a Circle by Archimedes

[8] Posamentier and Salkind, Challenging Problems in Geometry, Dover, 2nd edition, 1996: pp. 104–105, #4–23.

[9] College Mathematics Journal 29(4), September 1998, p. 331, problem 635.

[10] Johnson, Roger A., Advanced Euclidean Geometry, Dover Publ., 2007.

[11] Harkness, James (1898). Introduction to the theory of analytic functions. London, New York: Macmillan and Co. p. 30.

[12] Ogilvy, C. Stanley, Excursions in Geometry, Dover, 1969, 14–17.

[13] Altshiller-Court, Nathan, College Geometry, Dover, 2007 (orig. 1952).

[14] Incircle – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

[15] Circumcircle – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

[16] Tangential Polygon – from Wolfram MathWorld. Mathworld.wolfram.com (2012-04-26). Retrieved on 2012-05-03.

1.11 Further reading

• Pedoe, Dan (1988). Geometry: a comprehensive course. Dover.

• “Circle” in The MacTutor History of Mathematics archive

1.12 External links

• Hazewinkel, Michiel, ed. (2001), “Circle”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4 • Circle (PlanetMath.org website)

• Weisstein, Eric W., “Circle”, MathWorld. • Interactive Java applets for the properties of and elementary constructions involving circles.

• Interactive Standard Form Equation of Circle Click and drag points to see standard form equation in action • Munching on Circles at cut-the-knot

• Area of a Circle Calculate the basic properties of a circle. • MathAce’s Circle article – has a good in-depth explanation of unit circles and transforming circular equations.

• How to find the area of a circle. There are many types of problems involving how to find the area of circle; for example, finding area of a circle from its radius, diameter or circumference. Chapter 2

Conic section

Types of conic sections: 1. Parabola 2. Circle and ellipse 3. Hyperbola

In mathematics, a conic section (or just conic) is a curve obtained as the intersection of a cone (more precisely, a right circular conical surface) with a plane. In analytic geometry, a conic may be defined as a plane algebraic curve of degree 2, and as a quadric of dimension 1. There are a number of other geometric definitions possible. One of the most useful, in that it involves only the plane, is that a non-circular conic[1] consists of those points whose distances to some point, called a focus, and some line, called a directrix, are in a fixed ratio, called the eccentricity. Traditionally, the three types of conic section are the hyperbola, the parabola, and the ellipse. The circle is a special case of the ellipse, and is of sufficient interest in its own right that it is sometimes called the fourth type of conic section. The type of a conic corresponds to its eccentricity, those with eccentricity less than 1 being ellipses, those with eccentricity equal to 1 being parabolas, and those with eccentricity greater than 1 being hyperbolas. In the focus- directrix definition of a conic the circle is a limiting case with eccentricity 0. In modern geometry certain degenerate cases, such as the union of two lines, are included as conics as well.

16 2.1. HISTORY 17

The conic sections have been named and studied at least since 200 BC, when Apollonius of Perga undertook a systematic study of their properties.

2.1 History

2.1.1 Menaechmus and early works

It is believed that the first definition of a conic section is due to Menaechmus (died 320 BC). His work did not survive and is only known through secondary accounts. The definition used at that time differs from the one commonly used today in that it requires the plane cutting the cone to be perpendicular to one of the lines, (a generatrix), that generates the cone as a surface of revolution. Thus the shape of the conic is determined by the angle formed at the vertex of the cone (between two opposite generatrices): If the angle is acute then the conic is an ellipse; if the angle is right then the conic is a parabola; and if the angle is obtuse then the conic is a hyperbola. Note that the circle cannot be defined this way and was not considered a conic at this time. Euclid ( fl. 300 BC ) is said to have written four books on conics but these were lost as well.[2] Archimedes (died c. 212 BC) is known to have studied conics, having determined the area bounded by a parabola and an ellipse. The only part of this work to survive is a book on the solids of revolution of conics.

2.1.2 Apollonius of Perga

The greatest progress in the study of conics by the ancient Greeks is due to Apollonius of Perga (died c. 190 BC), whose eight-volume Conic Sections or Conics summarized and greatly extended existing knowledge. Apollonius’s major innovation was to characterize a conic using properties within the plane and intrinsic to the curve; this greatly simplified analysis. With this tool, it was now possible to show that any plane cutting the cone, regardless of its angle, will produce a conic according to the earlier definition, leading to the definition commonly used today. Pappus of Alexandria (died c. 350 CE) is credited with discovering the importance of the concept of a conic’s focus, and with the discovery of the related concept of a directrix.

2.1.3 Al-Kuhi

An instrument for drawing conic sections was ﬁrst described in 1000 CE by the Islamic mathematician Al-Kuhi.[3][4]

2.1.4 Omar Khayyám

Apollonius’s work was translated into Arabic (the technical language of the time) and much of his work only survives through the Arabic version. Persians found applications to the theory; the most notable of these was the Persian[5] mathematician and poet Omar Khayyám who used conic sections to solve algebraic equations.

2.1.5 Europe

Johannes Kepler extended the theory of conics through the "principle of continuity", a precursor to the concept of limits. Girard Desargues and Blaise Pascal developed a theory of conics using an early form of projective geometry and this helped to provide impetus for the study of this new ﬁeld. In particular, Pascal discovered a theorem known as the hexagrammum mysticum from which many other properties of conics can be deduced. Meanwhile, René Descartes applied his newly discovered Analytic geometry to the study of conics. This had the eﬀect of reducing the geometrical problems of conics to problems in algebra. 18 CHAPTER 2. CONIC SECTION

2.2 Features

The three types of conics are the ellipse, parabola, and hyperbola. The circle can be considered as a fourth type (as it was by Apollonius) or as a kind of ellipse. The circle and the ellipse arise when the intersection of cone and plane is a closed curve. The circle is obtained when the cutting plane is parallel to the plane of the generating circle of the cone – for a right cone as in the picture at the top of the page this means that the cutting plane is perpendicular to the symmetry axis of the cone. If the cutting plane is parallel to exactly one generating line of the cone, then the conic is unbounded and is called a parabola. In the remaining case, the ﬁgure is a hyperbola. In this case, the plane will intersect both halves (nappes) of the cone, producing two separate unbounded curves. Various parameters are associated with a conic section, as shown in the following table. (For the ellipse, the table gives the case of a>b, for which the major axis is horizontal; for the reverse case, interchange the symbols a and b. For the hyperbola the east-west opening case is given. In all cases, a and b are positive.) The non-circular conic sections are exactly those curves that, for a point F, a line L not containing F and a non-negative number e, are the locus of points whose distance to F equals e times their distance to L. F is called the focus, L the directrix, and e the eccentricity. The linear eccentricity (c) is the distance between the center and the focus (or one of the two foci). The latus rectum (2ℓ) is the chord parallel to the directrix and passing through the focus (or one of the two foci). The semi-latus rectum (ℓ) is half the latus rectum. The focal parameter (p) is the distance from the focus (or one of the two foci) to the directrix. The following relations hold:

• pe = ℓ

• ae = c.

2.3 Construction

There are many methods to construct a conic. One of them, that is useful in engineering applications, being parallelogram method, where a conic is constructed point by point by means of connecting certain equally spaced points on horizontal line and vertical line.

2.4 Properties

Just as two (distinct) points determine a line, five points determine a conic. Formally, given any five points in the plane in general linear position, meaning no three collinear, there is a unique conic passing through them, which will be non-degenerate; this is true over both the affine plane and projective plane. Indeed, given any five points there is a conic passing through them, but if three of the points are collinear the conic will be degenerate (reducible, because it contains a line), and may not be unique; see further discussion. Four points in the plane in general linear position determine a unique conic passing through the first three points and having the fourth point as its center. Thus knowing the center is equivalent to knowing two points on the conic for the purpose of determining the curve.[6]:p. 203 Furthermore, a conic is determined by any combination of k points in general position that it passes through and 5–k lines that are tangent to it, for 0≤k≤5.[7] Irreducible conic sections are always “smooth”. This is important for many applications, such as aerodynamics, where a smooth surface is required to ensure laminar flow and to prevent turbulence.

2.4.1 Intersection at inﬁnity

An algebro-geometrically intrinsic form of this classiﬁcation is by the intersection of the conic with the line at inﬁnity, which gives further insight into their geometry: 2.4. PROPERTIES 19

• ellipses intersect the line at infinity in 0 points—rather, in 0 real points, but in 2 complex points, which are conjugate; • parabolas intersect the line at infinity in 1 double point, corresponding to the axis—they are tangent to the line at infinity, and close at infinity, as distended ellipses; • hyperbolas intersect the line at infinity in 2 points, corresponding to the asymptotes—hyperbolas pass through infinity, with a twist. Going to infinity along one branch passes through the point at infinity corresponding to the asymptote, then re-emerges on the other branch at the other side but with the inside of the hyperbola (the direction of curvature) on the other side – left vs. right (corresponding to the non-orientability of the real projective plane)—and then passing through the other point at infinity returns to the first branch. Hyperbolas can thus be seen as ellipses that have been pulled through infinity and re-emerged on the other side, flipped.

2.4.2 Degenerate cases

For more details on this topic, see Degenerate conic.

There are five degenerate cases: three in which the plane passes through apex of the cone, and three that arise when the cone itself degenerates to a cylinder (a doubled line can occur in both cases). When the plane passes through the apex, the resulting conic is always degenerate, and is either: a point (when the angle between the plane and the axis of the cone is larger than tangential); a straight line (when the plane is tangential to the surface of the cone); or a pair of intersecting lines (when the angle is smaller than the tangential). These correspond respectively to degeneration of an ellipse, parabola, and a hyperbola, which are characterized in the same way by angle. The straight line is more precisely a double line (a line with multiplicity 2) because the plane is tangent to the cone, and thus the intersection should be counted twice. Where the cone is a cylinder, i.e. with the vertex at infinity, cylindric sections are obtained;[8] this corresponds to the apex being at infinity. Cylindrical sections are ellipses (or circles), unless the plane is vertical (which corresponds to passing through the apex at infinity), in which case three degenerate cases occur: two parallel lines, known as a ribbon (corresponding to an ellipse with one axis infinite and the other axis real and non-zero, the distance between the lines), a double line (an ellipse with one infinite axis and one axis zero), and no intersection (an ellipse with one infinite axis and the other axis imaginary).

2.4.3 Eccentricity, focus and directrix

The four defining conditions above can be combined into one condition that depends on a fixed point F (the focus), a line L (the directrix) not containing F and a nonnegative real number e (the eccentricity). The corresponding conic section consists of the locus of all points whose distance to F equals e times their distance to L . For 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola. For an ellipse and a hyperbola, two focus-directrix combinations can be taken, each giving the same full ellipse or hyperbola. The distance from the center to the directrix is a/e , where a is the semi-major axis of the ellipse, or the distance from the center to the tops of the hyperbola. The distance from the center to a focus is a e . The circle is a limiting case and is not defined by a focus and directrix in the plane. However, if one were to consider the directrix to be infinitely far from the center (the line at infinity), then by taking the eccentricity to be e = 0 a circle will have the focus-directrix property, but is still not defined by that property.[9] One must be careful in this situation to correctly use the definition of eccentricity as the ratio of the distance of a point on the circle to the focus (length of a radius) to the distance of that point to the directrix (this distance is infinite) which gives the limiting value of zero. The eccentricity of a conic section is thus a measure of how far it deviates from being circular. For a given a , the closer e is to 1, the smaller is the semi-minor axis.

2.4.4 Generalizations

Conics may be defined over other fields, and may also be classified in the projective plane rather than in the affine plane. 20 CHAPTER 2. CONIC SECTION

Over the complex numbers ellipses and hyperbolas are not distinct, since −1 is a square; precisely, the ellipse x2+y2 = 1 becomes a hyperbola under the substitution y = iw, geometrically a complex rotation, yielding x2 − w2 = 1 – a hyperbola is simply an ellipse with an imaginary axis length. Thus there is a 2-way classification: ellipse/hyperbola and parabola. Geometrically, this corresponds to intersecting the line at infinity in either 2 distinct points (corresponding to two asymptotes) or in 1 double point (corresponding to the axis of a parabola), and thus the real hyperbola is a more suggestive image for the complex ellipse/hyperbola, as it also has 2 (real) intersections with the line at infinity. In projective space, over any division ring, but in particular over either the real or complex numbers, all non-degenerate conics are equivalent, and thus in projective geometry one simply speaks of “a conic” without specifying a type, as type is not meaningful. Geometrically, the line at infinity is no longer special (distinguished), so while some conics intersect the line at infinity differently, this can be changed by a projective transformation – pulling an ellipse out to infinity or pushing a parabola off infinity to an ellipse or a hyperbola. A generalization of a non degenerate conic in a projective plane is an oval. An oval is a point set that has the following properties, which are held by conics: 1) any line intersects an oval in none, one or two points, 2) at any point of the oval there exists a unique tangent line.

2.4.5 In other areas of mathematics

The classification into elliptic, parabolic, and hyperbolic is pervasive in mathematics, and often divides a field into sharply distinct subfields. The classification mostly arises due to the presence of a quadratic form (in two variables this corresponds to the associated discriminant), but can also correspond to eccentricity. Quadratic form classifications: quadratic forms Quadratic forms over the reals are classified by Sylvester’s law of inertia, namely by their positive index, zero index, and negative index: a quadratic form in n variables can be converted to a diagonal form, 2 2 ··· 2 − 2 − · · · − 2 as x1 + x2 + + xk xk+1 xk+l, where the number of +1 coefficients, k, is the positive index, the number of −1 coefficients, l, is the negative index, and the remaining variables are the zero index m, so k + l + m = n. In two variables the non-zero quadratic forms are classified as:

• x2 + y2 – positive-deﬁnite (the negative is also included), corresponding to ellipses, • x2 – degenerate, corresponding to parabolas, and • x2 − y2 – indeﬁnite, corresponding to hyperbolas.

In two variables quadratic forms are classified by discriminant, analogously to conics, but in higher dimensions the more useful classification is as definite, (all positive or all negative), degenerate, (some zeros), or indefinite (mix of positive and negative but no zeros). This classification underlies many that follow. curvature The Gaussian curvature of a surface describes the infinitesimal geometry, and may at each point be either positive – elliptic geometry, zero – Euclidean geometry (flat, parabola), or negative – hyperbolic geometry; infinitesimally, to second order the surface looks like the graph of x2 + y2, x2 (or 0), or x2 − y2 . Indeed, by the uniformization theorem every surface can be taken to be globally (at every point) positively curved, flat, or negatively curved. In higher dimensions the Riemann curvature tensor is a more complicated object, but manifolds with constant sectional curvature are interesting objects of study, and have strikingly different properties, as discussed at sectional curvature.

Second order PDEs Partial differential equations (PDEs) of second order are classified at each point as elliptic, parabolic, or hyperbolic, accordingly as their second order terms correspond to an elliptic, parabolic, or hyperbolic quadratic form. The behavior and theory of these different types of PDEs are strikingly different – representative examples is that the Poisson equation is elliptic, the heat equation is parabolic, and the wave equation is hyperbolic.

Eccentricity classiﬁcations include:

Möbius transformations Real Möbius transformations (elements of PSL2(R) or its 2-fold cover, SL2(R)) are classiﬁed as elliptic, parabolic, or hyperbolic accordingly as their half-trace is 0 ≤ | tr |/2 < 1, | tr |/2 = 1, or | tr |/2 > 1, mirroring the classiﬁcation by eccentricity. 2.5. CARTESIAN COORDINATES 21

Variance-to-mean ratio The variance-to-mean ratio classiﬁes several important families of discrete probability distributions: the constant distribution as circular (eccentricity 0), binomial distributions as elliptical, Poisson distributions as parabolic, and negative binomial distributions as hyperbolic. This is elaborated at cumulants of some discrete probability distributions.

2.5 Cartesian coordinates

In the Cartesian coordinate system, the graph of a quadratic equation in two variables is always a conic section (though it may be degenerate), and all conic sections arise in this way. The equation will be of the form

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 with A, B, Czero. all not [10]

As scaling all six constants yields the same locus of zeros, one can consider conics as points in the ﬁve-dimensional projective space P5.

2.5.1 Discriminant classiﬁcation

The conic sections described by this equation can be classiﬁed with the discriminant[11]

B2 − 4AC.

If the conic is non-degenerate, then:

• if B2 − 4AC < 0 , the equation represents an ellipse;

• if A = C and B = 0 , the equation represents a circle, which is a special case of an ellipse;

• if B2 − 4AC = 0 , the equation represents a parabola;

• if B2 − 4AC > 0 , the equation represents a hyperbola;[12]

• if we also have A + C = 0 , the equation represents a rectangular hyperbola.

To distinguish the degenerate cases from the non-degenerate cases, let ∆ be the determinant of the 3×3 matrix [A, B/2, D/2 ; B/2, C, E/2 ; D/2, E/2, F ]: that is, ∆ = (AC - B2/4)F + BED/4 - CD2/4 - AE2/4. Then the conic section is non-degenerate if and only if ∆ ≠ 0. If ∆=0 we have a point ellipse, two parallel lines (possibly coinciding with each other) in the case of a parabola, or two intersecting lines in the case of a hyperbola.[13]:p.63 Moreover, in the case of a non-degenerate ellipse (with B2 − 4AC < 0 and ∆≠0), we have a real ellipse if C∆ < 0 but an imaginary ellipse if C∆ > 0. An example is x2 + y2 + 10 = 0 , which has no real-valued solutions. Note that A and B are polynomial coeﬃcients, not the lengths of semi-major/minor axis as deﬁned in some sources.

2.5.2 Matrix notation

Main article: Matrix representation of conic sections

The above equation can be written in matrix notation as

[ ] [ ] [ ] AB/2 x x y . . + Dx + Ey + F = 0. B/2 C y

The type of conic section is solely determined by the determinant of middle matrix: if it is positive, zero, or negative then the conic is an ellipse, parabola, or hyperbola respectively (see geometric meaning of a quadratic form). If both 22 CHAPTER 2. CONIC SECTION

the eigenvalues of the middle matrix are non-zero (i.e. it is an ellipse or a hyperbola), we can do a transformation of variables to obtain ( ) ( )( ) − T B − x a A 2 x a − B − = G y c 2 C y c where a,c, and G satisfy D + 2aA + Bc = 0,E + 2Cc + Ba = 0, and G = Aa2 + Cc2 + Bac − F . The quadratic can also be written as

    [ ] AB/2 D/2 x x y 1 .B/2 CE/2.y = 0. D/2 E/2 F 1 If the determinant of this 3×3 matrix is non-zero, the conic section is not degenerate. If the determinant equals zero, the conic is a degenerate parabola (two parallel or coinciding lines), a degenerate ellipse (a point ellipse), or a degenerate hyperbola (two intersecting lines). Note that in the centered equation with constant term G, G equals minus one times the ratio of the 3×3 determinant to the 2×2 determinant.

2.5.3 As slice of quadratic form

The equation

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0 can be rearranged by taking the aﬃne linear part to the other side, yielding

Ax2 + Bxy + Cy2 = −(Dx + Ey + F ). In this form, a conic section is realized exactly as the intersection of the graph of the quadratic form z = Ax2 + Bxy + Cy2 and the plane z = −(Dx + Ey + F ). Parabolas and hyperbolas can be realized by a horizontal plane ( D = E = 0 ), while ellipses require that the plane be slanted. Degenerate conics correspond to degenerate intersections, such as taking slices such as z = −1 of a positive-deﬁnite form.

2.5.4 Eccentricity in terms of parameters of the quadratic form

When the conic section is written algebraically as

Ax2 + Bxy + Cy2 + Dx + Ey + F = 0, the eccentricity can be written as a function of the parameters of the quadratic equation.[14] If 4AC = B2 the conic is a parabola and its eccentricity equals 1 (if it is non-degenerate). Otherwise, assuming the equation represents either a non-degenerate hyperbola or a non-degenerate, non-imaginary ellipse, the eccentricity is given by

√ √ 2 (A − C)2 + B2 e = √ , η(A + C) + (A − C)2 + B2 where η = 1 if the determinant of the 3×3 matrix is negative and η = −1 if that determinant is positive.

2.5.5 Standard form

Through a change of coordinates (a rotation of axes and a translation of axes) these equations can be put into standard forms:[15] 2.5. CARTESIAN COORDINATES 23

• Circle: x2 + y2 = a2 • Ellipse: x2/a2 + y2/b2 = 1 • Parabola: y2 = 4ax, x2 = 4ay • Hyperbola: x2/a2 − y2/b2 = 1, x2/b2 − y2/a2 = −1[16] • Rectangular hyperbola: xy = c2

The ﬁrst four of these forms are symmetric about both the x-axis and y-axis (for the circle, ellipse and hyperbola), or about either but not both (for the parabola). The rectangular hyperbola, however, is instead symmetric about the lines y = x and y = −x. These standard forms can be written as parametric equations,

• Circle:(a cos θ, a sin θ), • Ellipse:(a cos θ, b sin θ), • Parabola:(at2, 2at), • Hyperbola:(a sec θ, b tan θ) or (±a cosh u, b sinh u), • Rectangular hyperbola:(ct , c/t).

2.5.6 Invariants of conics [ ] AB/2 The trace and determinant of are both invariant with respect to both rotation of axes and translation B/2 C of the plane (movement of the origin).[12][17] The constant term F is invariant under rotation only.

2.5.7 Modiﬁed form

For some practical applications, it is important to re-arrange the standard form so that the focal-point can be placed at the origin. The mathematical formulation for a general conic section, with the other focus if any placed at a positive value (for an ellipse) or a negative value (for a hyperbola) on the horizontal axis, is then given in the polar form by

l r = 1 − e cos θ and in the Cartesian form by

√ x2 + y2 = (l + ex) ( )2 ( ) − le 2 2 x − 2 1 − e y ⇒ 1 e + = 1 l l2 1−e2 ( ) le From the above equation, the linear eccentricity (c) is given by c = 1−e2 . From the general equations given above, diﬀerent conic sections can be represented as shown below:

• Circle: x2 + y2 = r2

√ 2 • (x− a2−b2) y2 Ellipse: a2 + b2 = 1 • Parabola: y2 = 4a (x + a)

√ 2 • (x+ a2+b2) − y2 Hyperbola: a2 b2 = 1 24 CHAPTER 2. CONIC SECTION

2.6 Homogeneous coordinates

In homogeneous coordinates a conic section can be represented as:

2 2 2 A1x + A2y + A3z + 2B1xy + 2B2xz + 2B3yz = 0.

Or in matrix notation

    [ ] A1 B1 B2 x     x y z . B1 A2 B3 . y = 0. B2 B3 A3 z   A1 B1 B2   The matrix M = B1 A2 B3 is called the matrix of the conic section. B2 B3 A3   A1 B1 B2   ∆ = det(M) = det B1 A2 B3 is called the determinant of the conic section. If Δ = 0 then the conic B2 B3 A3 section is said to be degenerate; this means that the conic section is either a union of two straight lines, a repeated line, a point or the empty set.     [ ] 1 0 0 x For example, the conic section x y z .0 −1 0.y = 0 reduces to the union of two lines: 0 0 0 z

{x2 − y2 = 0} = {(x + y)(x − y) = 0} = {x + y = 0} ∪ {x − y = 0}.

Similarly, a conic section sometimes reduces to a (single) repeated line:

{x2 + 2xy + y2 = 0} = {(x + y)2 = 0} = {x + y = 0} ∪ {x + y = 0} = {x + y = 0}. ([ ]) A B δ = det 1 1 is called the discriminant of the conic section. If δ = 0 then the conic section is a parabola, if B1 A2 δ < 0, it is an hyperbola and if δ > 0, it is an ellipse. A conic section is a circle if δ > 0 and A1 = A2 and B1 = 0, it is 2 an rectangular hyperbola if δ < 0 and A1 = −A2. It can be proven that in the complex projective plane CP two conic sections have four points in common (if one accounts for multiplicity), so there are never more than 4 intersection points and there is always one intersection point (possibilities: four distinct intersection points, two singular intersection points and one double intersection points, two double intersection points, one singular intersection point and 1 with multiplicity 3, 1 intersection point with multiplicity 4). If there exists at least one intersection point with multiplicity > 1, then the two conic sections are said to be tangent. If there is only one intersection point, which has multiplicity 4, the two conic sections are said to be osculating.[18] Furthermore each straight line intersects each conic section twice. If the intersection point is double, the line is said to be tangent and it is called the tangent line. Because every straight line intersects a conic section twice, each conic section has two points at infinity (the intersection points with the line at infinity). If these points are real, the conic section must be a hyperbola, if they are imaginary conjugated, the conic section must be an ellipse, if the conic section has one double point at infinity it is a parabola. If the points at infinity are (1,i,0) and (1,-i,0), the conic section is a circle (see circular points at infinity). If a conic section has one real and one imaginary point at infinity or it has two imaginary points that are not conjugated then it not a real conic section (its coefficients are complex).

2.7 Polar coordinates

In polar coordinates, a conic section with one focus at the origin and, if any, the other at a negative value (for an ellipse) or a positive value (for a hyperbola) on the x-axis, is given by the equation 2.8. PENCIL OF CONICS 25

l r = , 1 + e cos θ where e is the eccentricity and l is the semi-latus rectum (see above). As above, for e = 0, we have a circle, for 0 < e < 1 we obtain an ellipse, for e = 1 a parabola, and for e > 1 a hyperbola.

2.8 Pencil of conics

A (non-degenerate) conic is completely determined by five points in general position (no three collinear) in a plane and the system of conics which pass through a fixed set of four points (again in a plane and no three collinear) is called a pencil of conics.[19] The four common points are called the base points of the pencil. Through any point other than a base point, there passes a single conic of the pencil. This concept generalizes a pencil of circles. In a projective plane defined over an algebraically closed field any two conics meet in four points (counted with multiplicity) and so, determine the pencil of conics based on these four points. Furthermore, the four base points determine three line pairs (degenerate conics through the base points, each line of the pair containing exactly two base points) and so each pencil of conics will contain at most three degenerate conics.[20]

A pencil of conics can represented algebraically in the following way. Let C1 and C2 be two distinct conics in a projective plane deﬁned over an algebraically closed ﬁeld K. For every pair λ, μ of elements of K, not both zero, the expression:

λC1 + µC2

represents a conic in the pencil determined by C1 and C2. This symbolic representation can be made concrete with a slight abuse of notation (using the same notation to denote the object as well as the equation deﬁning the object.) Thinking of C1, say, as a ternary quadratic form, then C1 = 0 is the equation of the “conic C1". Another concrete realization would be obtained by thinking of C1 as the 3×3 symmetric matrix which represents it. If C1 and C2 have such concrete realizations then every member of the above pencil will as well. Since the setting uses homogeneous coordinates in a projective plane, two concrete representations (either equations or matrices) give the same conic if they diﬀer by a non-zero multiplicative constant.

2.9 Intersecting two conics

The solutions to a system of two second degree equations in two variables may be viewed as the coordinates of the points of intersection of two generic conic sections. In particular two conics may possess none, two or four possibly coincident intersection points. An eﬃcient method of locating these solutions exploits the homogeneous matrix representation of conic sections, i.e. a 3x3 symmetric matrix which depends on six parameters. The procedure to locate the intersection points follows these steps, where the conics are represented by matrices:

• given the two conics C1 and C2 , consider the pencil of conics given by their linear combination λC1 + µC2.

• identify the homogeneous parameters (λ, µ) which correspond to the degenerate conic of the pencil. This can be done by imposing the condition that det(λC1 + µC2) = 0 and solving for λ and µ . These turn out to be the solutions of a third degree equation.

• given the degenerate conic C0 , identify the two, possibly coincident, lines constituting it.

• intersect each identiﬁed line with either one of the two original conics; this step can be done eﬃciently using the dual conic representation of C0

• the points of intersection will represent the solutions to the initial equation system. 26 CHAPTER 2. CONIC SECTION

2.10 Applications

Conic sections are important in astronomy: the orbits of two massive objects that interact according to Newton’s law of universal gravitation are conic sections if their common center of mass is considered to be at rest. If they are bound together, they will both trace out ellipses; if they are moving apart, they will both follow parabolas or hyperbolas. See two-body problem. In projective geometry, the conic sections in the projective plane are equivalent to each other up to projective transformations. For speciﬁc applications of each type of conic section, see the articles circle, ellipse, parabola, and hyperbola. For certain fossils in paleontology, understanding conic sections can help understand the three-dimensional shape of certain organisms.

2.11 See also

• Circumconic and inconic

• Conic Sections Rebellion

• Dandelin spheres

• Director circle

• Elliptic coordinate system

• Focus (geometry), an overview of properties of conic sections related to the foci

• Lambert conformal conic projection

• Matrix representation of conic sections

• Nine-point conic

• Parabolic coordinates

• Projective conics

• Quadratic function

• Quadrics, the higher-dimensional analogs of conics

2.12 Notes

[1] Eves 1963, p. 319

[2] Heath, T.L., The Thirteen Books of Euclid’s Elements, Vol. I, Dover, 1956, pg.16

[3] Stillwell, John (2010). Mathematics and its history (3rd ed.). New York: Springer. p. 30. ISBN 1-4419-6052-X.

[4] “Apollonius of Perga Conics Books One to Seven” (PDF). Retrieved 10 June 2011.

[5] Turner, Howard R. (1997). Science in medieval Islam: an illustrated introduction. University of Texas Press. p. 53. ISBN 0-292-78149-0., Chapter , p. 53

[6] Whitworth, William Allen. Trilinear Coordinates and Other Methods of Modern Analytical Geometry of Two Dimen- sions, Forgotten Books, 2012 (orig. Deighton, Bell, and Co., 1866). http://www.forgottenbooks.com/search?q=Trilinear+ coordinates&t=books

[7] Paris Pamﬁlos, “A gallery of conics by ﬁve elements”, Forum Geometricorum 14, 2014, 295-−348. http://forumgeom.fau. edu/FG2014volume14/FG201431.pdf

[8] “MathWorld: Cylindric section”. 2.13. REFERENCES 27

[9] Eves 1963, p. 320

[10] Protter & Morrey (1970, p. 316)

[11] Fanchi, John R. (2006), Math refresher for scientists and engineers, John Wiley and Sons, pp. 44–45, ISBN 0-471-75715-2, Section 3.2, page 45

[12] Protter & Morrey (1970, p. 326)

[13] Lawrence, J. Dennis, A Catalog of Special Plane Curves, Dover Publ., 1972.

[14] Ayoub, Ayoub B., “The eccentricity of a conic section,” The College Mathematics Journal 34(2), March 2003, 116–121.

[15] Protter & Morrey (1970, pp. 314–328,585–589)

[16] Protter & Morrey (1970, pp. 290–314)

[17] Pettofrezzo, Anthony, Matrices and Transformations, Dover Publ., 1966, pp. 101-111.

[18] Wilczynski, E. J. (1916), “Some remarks on the historical development and the future prospects of the diﬀerential geometry of plane curves”, Bull. Amer. Math. Soc. 22: 317–329, doi:10.1090/s0002-9904-1916-02785-6.

[19] Faulkner 1952, pg. 64

[20] Samuel 1988, pg. 50

2.13 References

• Akopyan, A.V. and Zaslavsky, A.A. (2007). Geometry of Conics. American Mathematical Society. p. 134. ISBN 0-8218-4323-0.

• Eves, Howard (1963), A Survey of Geometry (Volume One), Boston: Allyn and Bacon

• Faulkner, T. E. (1952), Projective Geometry (2nd ed.), Edinburgh: Oliver and Boyd

• Protter, Murray H.; Morrey, Jr., Charles B. (1970), College Calculus with Analytic Geometry (2nd ed.), Reading: Addison-Wesley, LCCN 76087042

• Samuel, Pierre (1988), Projective Geometry, Undergraduate Texts in Mathematics (Readings in Mathematics), New York: Springer-Verlag, ISBN 0-387-96752-4

2.14 External links

• Derivations of Conic Sections at Convergence • Conic sections at Special plane curves. • Weisstein, Eric W., “Conic Section”, MathWorld. • Determinants and Conic Section Curves • Occurrence of the conics. Conics in nature and elsewhere. • Conics. An essay on conics and how they are generated. • See Conic Sections at cut-the-knot for a sharp proof that any ﬁnite conic section is an ellipse and Xah Lee for a similar treatment of other conics. • Cone-plane intersection MATLAB code • Eight Point Conic at Dynamic Geometry Sketches • An interactive Java conics grapher; uses a general second-order implicit equation. 28 CHAPTER 2. CONIC SECTION

Table of conics, Cyclopaedia, 1728 2.14. EXTERNAL LINKS 29

Diagram from Apollonius’ Conics, in a 9th century Arabic translation 30 CHAPTER 2. CONIC SECTION

Circle Ellipse Parabola Hyperbola

Conics are of three types: parabolas, ellipses, including circles, and hyperbolas. 2.14. EXTERNAL LINKS 31

directrices

focal parameter Latus rectum Minor axis Major axis linear eccentricity

foci

Conic parameters in the case of an ellipse 32 CHAPTER 2. CONIC SECTION

e=0.5 F M e=1

e=2

e=∞ M'

Ellipse (e=1/2), parabola (e=1) and hyperbola (e=2) with ﬁxed focus F and directrix (e=∞). 2.14. EXTERNAL LINKS 33

Standard forms of an ellipse 34 CHAPTER 2. CONIC SECTION

Standard forms of a parabola 2.14. EXTERNAL LINKS 35

Standard forms of a hyperbola 36 CHAPTER 2. CONIC SECTION

(a) (b)

(c)

Three diﬀerent types of conic sections. Focal-points corresponding to all conic sections are placed at the origin. 2.14. EXTERNAL LINKS 37

Development of the conic section as the eccentricity e increases 38 CHAPTER 2. CONIC SECTION

The paraboloid shape of Archeocyathids produces conic sections on rock faces Chapter 3

Focus (geometry)

Point F is a focus point for the red ellipse, green parabola and blue hyperbola.

In geometry, the foci (/ˈfoʊsaɪ/; singular focus) are a pair of special points with reference to which any of a variety of curves is constructed. For example, foci can be used in deﬁning conic sections, the four types of which are the

39 40 CHAPTER 3. FOCUS (GEOMETRY) circle, ellipse, parabola, and hyperbola. In addition, foci are used to deﬁne the Cassini oval and the Cartesian oval.

3.1 Conic sections

See also: Conic section § Eccentricity, focus and directrix, Ellipse § Focus, Parabola § Focal length and Hyperbola § Directrix and focus

3.1.1 Deﬁning conics in terms of two foci

An ellipse can be defined as the locus of points for each of which the sum of the distances to two given foci is a constant. A circle is the special case of an ellipse in which the two foci coincide with each other. Thus, a circle can be more simply defined as the locus of points each of which is a fixed distance from a single given focus. A circle can also be defined as the circle of Apollonius, in terms of two different foci, as the set of points having a fixed ratio of distances to the two foci. A parabola is a limiting case of an ellipse in which one of the foci is a point at infinity. A hyperbola can be defined as the locus of points for each of which the absolute value of the difference between the distances to two given foci is a constant.

3.1.2 Deﬁning conics in terms of a focus and a directrix

It is also possible to describe all the conic sections in terms of a single focus and a single directrix, which is a given line not containing the focus. A conic is defined as the locus of points for each of which the distance to the focus divided by the distance to the directrix is a fixed positive constant, called the eccentricity e. If e is between zero and one the conic is an ellipse; if e=1 the conic is a parabola; and if e>1 the conic is a hyperbola. If the distance to the focus is fixed and the directrix is a line at infinity, so the eccentricity is zero, then the conic is a circle.

3.1.3 Deﬁning conics in terms of a focus and a directrix circle

It is also possible to describe all the conic sections as loci of points that are equidistant from a single focus and a single, circular directrix. For the ellipse, both the focus and the center of the directrix circle have finite coordinates and the radius of the directrix circle is greater than the distance between the center of this circle and the focus; thus, the focus is inside the directrix circle. The ellipse thus generated has its second focus at the center of the directrix circle, and the ellipse lies entirely within the circle. For the parabola, the center of the directrix moves to the point at infinity (see projective geometry). The directrix 'circle' becomes a curve with zero curvature, indistinguishable from a straight line. The two arms of the parabola become increasingly parallel as they extend, and 'at infinity' become parallel; using the principles of projective geometry, the two parallels intersect at the point at infinity and the parabola becomes a closed curve (elliptical projection). To generate a hyperbola, the radius of the directrix circle is chosen to be less than the distance between the center of this circle and the focus; thus, the focus is outside the directrix circle. The arms of the hyperbola approach asymptotic lines and the 'right-hand' arm of one branch of a hyperbola meets the 'left-hand' arm of the other branch of a hyperbola at the point at infinity; this is based on the principle that, in projective geometry, a single line meets itself at a point at infinity. The two branches of a hyperbola are thus the two (twisted) halves of a curve closed over infinity. In projective geometry, all conics are equivalent in the sense that every theorem that can be stated for one can be stated for the others.

3.1.4 Astronomical signiﬁcance

See also: Ellipse § Planetary orbits 3.2. CARTESIAN AND CASSINI OVALS 41

In the gravitational two-body problem, the orbits of the two bodies about each other are described by two overlapping conic sections with one of the foci of one being coincident with one of the foci of the other at the center of mass (barycenter) of the two bodies. Thus, for instance, the minor planet Pluto's largest moon Charon has an elliptical orbit which has one focus at the Pluto-Charon system’s barycenter, which is a point that is in space between the two bodies; and Pluto also moves in an ellipse with one of its foci at that same barycenter between the bodies. Pluto’s ellipse is entirely inside Charon’s ellipse, as shown in this animation of the system. By comparison, the Earth’s Moon moves in an ellipse with one of its foci at the barycenter of the Moon and the Earth, this barycenter being within the Earth itself, while the Earth (more precisely, its center) moves in an ellipse with one focus at that same barycenter within the Earth. The barycenter is about three-quarters of the distance from Earth’s center to its surface. Moreover, the Pluto-Charon system moves in an ellipse around its barycenter with the Sun, as does the Earth-Moon system. In both cases the barycenter is well within the body of the Sun. Two binary stars also move in ellipses sharing a focus at their barycenter; for an animation, see here.

3.2 Cartesian and Cassini ovals

A Cartesian oval is the set of points for each of which the weighted sum of the distances to two given foci is a constant. If the weights are equal, the special case of an ellipse results. A Cassini oval is the set of points for each of which the product of the distances to two given foci is a constant.

3.3 Generalization

The concept of a focus can be generalized to arbitrary algebraic curves. Let C be a curve of class m and let I and J denote the circular points at infinity. Draw the m tangents to C through each of I and J. There are two sets of m lines which will have m2 points of intersection, with exceptions in some cases due to singularities, etc. These points of intersection are the defined to be the foci of C. In other words, a point P is a focus if both PI and PJ are tangent to C. When C is a real curve, only the intersections of conjugate pairs are real, so there are m in a real foci and m2−m imaginary foci. When C is a conic, the real foci defined this way are exactly the foci which can be used in the geometric construction of C.

3.4 Confocal curves

Let P1, P2, …, Pm be given as foci of a curve C of class m. Let P be the product of the tangential equations of these points and Q the product of the tangential equations of the circular points at inﬁnity. Then all the lines which are common tangents to both P=0 and Q=0 are tangent to C. So, by the AF+BG theorem, the tangential equation of C has the form HP+KQ=0. Since C has class m, H must be a constant and K but have degree less than or equal to m−2. The case H=0 can be eliminated as degenerate, so the tangential equation of C can be written as P+fQ=0 where f is an arbitrary polynomial of degree m−2.[1] 2 For example, let P1=(1,0), P2=(−1,0). The tangential equations are X+1=0 and X−1=0 so P= X −1=0. The tangential equations for the circular points at inﬁnity are X+iY=0 and X−iY=0 so Q=X2+Y2. Therefore the tangential equation for a conic with the given foci is X2−1+c(X2+Y2)=0, or (1+c)X2+cY2=1 where c is an arbitrary constant. In point coordinates this becomes

x2 y2 + = 1. 1 + c c

3.5 References

[1] Follows Hilton p. 69 with an appeal to AF+BG for simpliﬁcation. 42 CHAPTER 3. FOCUS (GEOMETRY)

• Hilton, Harold (1920). Plane Algebraic Curves. Oxford. p. 69. Chapter 4

Locus (mathematics)

Each curve in this example is a locus deﬁned as the conchoid of a circle centered at point P and the line l. In this example, P is 7cm from l.

In geometry, a locus (plural: loci) is a set of points whose location satisﬁes or is determined by one or more speciﬁed conditions.[1][2]

4.1 Commonly studied loci

Examples from plane geometry include:

• The set of points equidistant from two points is a perpendicular bisector to the line segment connecting the two points.[3]

• The set of points equidistant from two lines which cross is the angle bisector.

• All conic sections are loci:[4]

• Parabola: the set of points equidistant from a single point (the focus) and a line (the directrix). • Circle: the set of points for which the distance from a single point is constant (the radius). The set of points for each of which the ratio of the distances to two given foci is a positive constant (that is not 1) is referred to as a Circle of Apollonius. • Hyperbola: the set of points for each of which the absolute value of the diﬀerence between the distances to two given foci is a constant. • Ellipse: the set of points for each of which the sum of the distances to two given foci is a constant. The circle is the special case in which the two foci coincide with each other.

43 44 CHAPTER 4. LOCUS (MATHEMATICS)

4.2 Proof of a locus

In order to prove that a geometric shape is the correct locus for a given set of conditions, one generally divides the proof into two stages:[5]

• Proof that all the points that satisfy the conditions are on the given shape. • Proof that all the points on the given shape satisfy the conditions.

4.3 Examples

P y

A x

(distance PA) = 3.(distance PB)

4.3.1 First example

We ﬁnd the locus of the points P that have a given ratio of distances k = d1/d2 to two given points. In this example we choose k= 3, A(−1,0) and B(0,2) as the ﬁxed points. 4.3. EXAMPLES 45

P(x,y) is a point of the locus ⇔ |PA| = 3|PB| ⇔ |PA|2 = 9|PB|2 ⇔ (x + 1)2 + (y − 0)2 = 9(x − 0)2 + 9(y − 2)2 ⇔ 8(x2 + y2) − 2x − 36y + 35 = 0 ( ) ( ) ⇔ − 1 2 − 9 2 45 x 8 + y 4 = 64 √ 3 This equation represents a circle with center (1/8,9/4) and radius 8 5 . It is the circle of Apollonius deﬁned by these values of k, A, and B.

4.3.2 Second example

C y

A O B x

Locus of point C

A triangle ABC has a ﬁxed side [AB] with length c. We determine the locus of the third vertex C such that the medians from A and C are orthogonal. We choose an orthonormal coordinate system such that A(-c/2,0), B(c/2,0). C(x,y) is the variable third vertex. The center of [BC] is M( (2x+c)/4, y/2 ). The median from C has a slope y/x. The median AM has slope 2y/(2x+3c).

C(x,y) is a point of the locus ⇔ The medians from A and C are orthogonal ⇔ y · 2y − x 2x+3c = 1 ⇔ 2y2 + 2x2 + 3cx = 0 ⇔ x2 + y2 + (3c/2)x = 0 ⇔ (x + 3c/4)2 + y2 = 9c2/16

The locus of the vertex C is a circle with center (−3c/4,0) and radius 3c/4. 46 CHAPTER 4. LOCUS (MATHEMATICS)

C y

A O B x

The locus is a circle

4.3.3 Third example

A locus can also be defined by two associated curves depending on one common parameter. If the parameter varies, the intersection points of the associated curves describe the locus. In the figure, the points K and L are fixed points on a given line m. The line k is a variable line through K. The line l through L is perpendicular to k. The angle α between k and m is the parameter. k and l are associated lines depending on the common parameter. The variable intersection point S of k and l describes a circle. This circle is the locus of the intersection point of the two associated lines.

4.3.4 Fourth example

A locus of points need not be one-dimensional (as a circle, line, etc.). For example,[1] the locus of the inequality 2x+3y–6<0 is the portion of the plane that is below the line 2x+3y–6=0.

4.4 See also

• Line (geometry) • Curve

4.5 References

[1] James, Robert Clarke; James, Glenn (1992), Mathematics Dictionary, Springer, p. 255, ISBN 978-0-412-99041-0

[2] Whitehead, Alfred North (1911), An Introduction to Mathematics, H. Holt, p. 121, ISBN 978-1-103-19784-2

[3] George E. Martin, The Foundations of Geometry and the Non-Euclidean Plane, Springer-Verlag, 1975 4.5. REFERENCES 47

l k S

K a L m

The intersection point of the associated lines k and l describes the circle

[4] Hamilton, Henry Parr (834), An Analytical System of Conic Sections: Designed for the Use of Students, Springer

[5] G.P. WestThe new geometry: form 1 Chapter 5

Point (geometry)

In modern mathematics, a point refers usually to an element of some set called a space. More specifically, in Euclidean geometry, a point is a primitive notion upon which the geometry is built. Being a primitive notion means that a point cannot be defined in terms of previously defined objects. That is, a point is defined only by some properties, called axioms, that it must satisfy. In particular, the geometric points do not have any length, area, volume, or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space.

5.1 Points in Euclidean geometry

Points, considered within the framework of Euclidean geometry, are one of the most fundamental objects. Euclid originally defined the point as “that which has no part”. In two-dimensional Euclidean space, a point is represented by an ordered pair (x, y) of numbers, where the first number conventionally represents the horizontal and is often denoted by x, and the second number conventionally represents the vertical and is often denoted by y. This idea is easily generalized to three-dimensional Euclidean space, where a point is represented by an ordered triplet (x, y, z) with the additional third number representing depth and often denoted by z. Further generalizations are represented by an ordered tuplet of n terms, (a1, a2,…, an) where n is the dimension of the space in which the point is located. Many constructs within Euclidean geometry consist of an infinite collection of points that conform to certain axioms. This is usually represented by a set of points; As an example, a line is an infinite set of points of the form

L={(a1,a2,...an)|a1c1+a2c2+...ancn=d} , where c1 through cn and d are constants and n is the dimension of the space. Similar constructions exist that define the plane, line segment and other related concepts. By the way, a degenerate line segment consists of only one point. In addition to defining points and constructs related to points, Euclid also postulated a key idea about points; he claimed that any two points can be connected by a straight line. This is easily confirmed under modern expansions of Euclidean geometry, and had lasting consequences at its introduction, allowing the construction of almost all the geometric concepts of the time. However, Euclid’s postulation of points was neither complete nor definitive, as he occasionally assumed facts about points that did not follow directly from his axioms, such as the ordering of points on the line or the existence of specific points. In spite of this, modern expansions of the system serve to remove these assumptions.

5.2 Dimension of a point

There are several inequivalent deﬁnitions of dimension in mathematics. In all of the common deﬁnitions, a point is 0-dimensional.

48 5.2. DIMENSION OF A POINT 49

-2 -1 1 2

-1

-2

A ﬁnite set of points (blue) in two-dimensional Euclidean space.

5.2.1 Vector space dimension

Main article: Dimension (vector space)

The dimension of a vector space is the maximum size of a linearly independent subset. In a vector space consisting of a single point (which must be the zero vector 0), there is no linearly independent subset. The zero vector is not itself linearly independent, because there is a non trivial linear combination making it zero: 1 · 0 = 0 .

5.2.2 Topological dimension

Main article: Lebesgue covering dimension

The topological dimension of a topological space X is defined to be the minimum value of n, such that every finite open cover A of X admits a finite open cover B of X which refines A in which no point is included in more than n+1 elements. If no such minimal n exists, the space is said to be of infinite covering dimension. A point is zero-dimensional with respect to the covering dimension because every open cover of the space has a refinement consisting of a single open set. 50 CHAPTER 5. POINT (GEOMETRY)

5.2.3 Hausdorﬀ dimension

Let X be a metric space. If S ⊂ X and d ∈ [0, ∞), the d-dimensional Hausdorff content of S is the infimum of the { ∈ } set of numbers δ ≥ 0 such∑ that there is some (indexed) collection of balls B(xi, ri): i I covering S with ri > 0 d for each i ∈ I that satisfies i∈I ri < δ . The Hausdorff dimension of X is defined by

{ ≥ d } dimH(X) := inf d 0 : CH (X) = 0 .

A point has Hausdorﬀ dimension 0 because it can be covered by a single ball of arbitrarily small radius.

5.3 Geometry without points

Although the notion of a point is generally considered fundamental in mainstream geometry and topology, there are some systems that forgo it, e.g. noncommutative geometry and pointless topology. A “pointless” or “pointfree” space is deﬁned not as a set, but via some structure (algebraic or logical respectively) which looks like a well-known function space on the set: an algebra of continuous functions or an algebra of sets respectively. More precisely, such structures generalize well-known spaces of functions in a way that the operation “take a value at this point” may not be deﬁned. A further tradition starts from some books of A. N. Whitehead in which the notion of region is assumed as a primitive together with the one of inclusion or connection.

5.4 Point masses and the Dirac delta function

Main article: Dirac delta function

Often in physics and mathematics, it is useful to think of a point as having non-zero mass or charge (this is especially common in classical electromagnetism, where electrons are idealized as points with non-zero charge). The Dirac delta function, or δ function, is (informally) a generalized function on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.[1][2][3] The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized point mass or point charge.[4] It was introduced by theoretical physicist Paul Dirac. In the context of signal processing it is often referred to as the unit impulse symbol (or function).[5] Its discrete analog is the Kronecker delta function which is usually defined on a finite domain and takes values 0 and 1.

5.5 See also

• Accumulation point

• Aﬃne space

• Boundary point

• Critical point

• Cusp

• Foundations of geometry

• Position (geometry)

• Pointwise

• Singular point of a curve 5.6. REFERENCES 51

5.6 References

[1] Dirac 1958, §15 The δ function, p. 58

[2] Gel'fand & Shilov 1968, Volume I, §§1.1, 1.3

[3] Schwartz 1950, p. 3

[4] Arfken & Weber 2000, p. 84

[5] Bracewell 1986, Chapter 5

• Clarke, Bowman, 1985, "Individuals and Points," Notre Dame Journal of Formal Logic 26: 61–75.

• De Laguna, T., 1922, “Point, line and surface as sets of solids,” The Journal of Philosophy 19: 449–61. • Gerla, G., 1995, "Pointless Geometries" in Buekenhout, F., Kantor, W. eds., Handbook of incidence geometry: buildings and foundations. North-Holland: 1015–31. • Whitehead A. N., 1919. An Enquiry Concerning the Principles of Natural Knowledge. Cambridge Univ. Press. 2nd ed., 1925. • ------, 1920. The Concept of Nature. Cambridge Univ. Press. 2004 paperback, Prometheus Books. Being the 1919 Tarner Lectures delivered at Trinity College.

• ------, 1979 (1929). Process and Reality. Free Press.

5.7 External links

• Deﬁnition of Point with interactive applet

• Points deﬁnition pages, with interactive animations that are also useful in a classroom setting. Math Open Reference

• Point at PlanetMath.org. • Weisstein, Eric W., “Point”, MathWorld. 52 CHAPTER 5. POINT (GEOMETRY)

5.8 Text and image sources, contributors, and licenses

5.8.1 Text

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and Anonymous: 1063 • Conic section Source: https://en.wikipedia.org/wiki/Conic_section?oldid=674649878 Contributors: AxelBoldt, Zundark, Tarquin, An- dre Engels, Danny, XJaM, Enchanter, Merphant, Hfastedge, Lir, Patrick, Michael Hardy, Dominus, Wapcaplet, Eric119, Looxix~enwiki, Ellywa, AugPi, Caramdir~enwiki, Raven in Orbit, Ideyal, EL Willy, Charles Matthews, Dcoetzee, Dino, Lumos3, Robbot, Rvollmert, Sverdrup, Academic Challenger, Tobias Bergemann, Tosha, Giftlite, Dbenbenn, Gene Ward Smith, Paisa, BenFrantzDale, Herbee, Frop- uff, Peruvianllama, Wwoods, Yekrats, Jorge Stolfi, Knutux, Quadell, Rdsmith4, Woofles, Halo, Almit39, Nickptar, Urhixidur, ELApro, Klaas van Aarsen, Discospinster, Mike Capp, Sam Derbyshire, Paul August, DcoetzeeBot~enwiki, Bender235, Brian0918, BenjBot, Rgdboer, Army1987, Duk, Ncurses, Nk, Knucmo2, Jumbuck, Arthena, Swift, Nick.sideras, Cmapm, Oleg Alexandrov, Nuno Tavares, OwenX, Woohookitty, LOL, StradivariusTV, Christopher Thomas, Asdfdsa, Rjwilmsi, Salix 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Tran Quoc123, Ptbotgourou, Senator Palpatine, Vincnet, AnomieBOT, ImperatorExercitus, Xqbot, BasilRazi, Blondandy, Srinivas.zinka, Rb88guy, GhalyBot, Geometryfan, NoldorinElf, FrescoBot, Barsamin, Tkuvho, RedBot, SpaceFlight89, Hager e, Rausch, Grabba, January, JelmerW, Andrw1491, Duoduoduo, EmausBot, John of Reading, GoingBatty, Mmei- jeri, Wmayner, Bomazi, Whoop whoop pull up, Olimparis, Anita5192, Whommighter, Modelpanicer, Pushhigher, Wcherowi, Helpful Pixie Bot, BG19bot, Mrjohncummings, BattyBot, Kelvinsong, ChengduTeacher, Ag2gaeh, EFZR090440, Wamiq, Loraof, KasparBot and Anonymous: 182 • Focus (geometry) Source: https://en.wikipedia.org/wiki/Focus_(geometry)?oldid=651192995 Contributors: Michael Hardy, Darkwind, Robbot, Jyril, Fropuff, Pmanderson, Kwamikagami, Wood Thrush, Woohookitty, SeventyThree, Mike Segal, Pfctdayelise, RobotE, RD- Bury, Melchoir, Metacomet, Hgilbert, TwistOfCain, Amakuru, Daniel5127, Herd of Swine, Futurebird, Deflective, Aliazimi, Brogand- man, VolkovBot, BotKung, Darrenlorent, ClueBot, Marino-slo, Addbot, AkhtaBot, Download, SpBot, Zorrobot, Fryed-peach, Luckas- bot, Xqbot, PrometheusDesmotes, DixonDBot, Duoduoduo, EmausBot, ZéroBot, Wayne Slam, Petrb, Tagremover, Loraof and Anony- mous: 19 • Locus (mathematics) Source: https://en.wikipedia.org/wiki/Locus_(mathematics)?oldid=650586099 Contributors: XJaM, Patrick, Michael Hardy, Robbot, Tomchiukc, Stewartadcock, Adam78, Tosha, Giftlite, Fudoreaper, Macrakis, Iamunknown, Maurreen, Alansohn, Joolz, Mac Davis, Snowolf, Ceyockey, Oleg Alexandrov, Linas, Dionyziz, SeventyThree, Seidenstud, Salix alba, NeonMerlin, FlaBot, Cia- Pan, YurikBot, Ninly, RDBury, Unschool, Adam majewski, Gilliam, JCSantos, Octahedron80, DHN-bot~enwiki, Wine Guy, Raptur, Gregorydavid, Mets501, Vanisaac, Sakurambo, CBM, Doctormatt, Peripitus, Omicronpersei8, Gnewf, Thijs!bot, Pjvpjv, Décartes, Fu- turebird, VoABot II, David Eppstein, Geboy, Viraaj456, BKred, Funandtrvl, VolkovBot, Anonymous Dissident, Kmhkmh, AlleborgoBot, SieBot, AkvoD3, Faradayplank, Spartan-James, Denisarona, ClueBot, Methossant, The Thing That Should Not Be, Daniel Hershcovich, DragonBot, HexaChord, Addbot, AkhtaBot, Vishnava, Luckas-bot, Yobot, KamikazeBot, AnomieBOT, Choij, JackieBot, Bepa~enwiki, Xqbot, FactCheckRedux, Erik9bot, Venkat.athma, Yghwtrrl, Wa03, Nethis, Jujutacular, Tacirupeca zula, Vrenator, Grace Xu, Josve05a, Érico Júnior Wouters, D.Lazard, Chewings72, Cgt, ResearchRave, ClueBot NG, CocuBot, Satellizer, Widr, Helpful Pixie Bot, Trump- kinius, Atomician, Qetuth, Mavi17, YFdyh-bot, Brirush, ShaneBWH, TheBlueCanoe, SkylonS, UGoodman, Surya939, Jhncls, Retartist, Nikhil Reddy007, Loraof and Anonymous: 106 • Point (geometry) Source: https://en.wikipedia.org/wiki/Point_(geometry)?oldid=670007931 Contributors: The Anome, William Avery, Michael Hardy, Fred Bauder, Aarchiba, Andres, Revolver, Charles Matthews, Dysprosia, Maximus Rex, Furrykef, Robbot, Kuszi, Giftlite, BenFrantzDale, MrMambo, Knutux, LiDaobing, Antandrus, Joseph Myers, Pmanderson, Neutrality, Freakofnurture, Discospinster, Paul August, Elwikipedista~enwiki, Rgdboer, Bobo192, Smalljim, PWilkinson, Sebastian Goll, Alansohn, Oleg Alexandrov, Linas, Mad- mardigan53, Bkkbrad, Unixer, Hdante, MassGalactusUniversum, Magister Mathematicae, Jobnikon, Tlroche, Salix alba, MikeJ9919, Sango123, Ravidreams, FlaBot, Chobot, Roboto de Ajvol, YurikBot, Wavelength, Trovatore, Bota47, Lt-wiki-bot, Arthur Rubin, Pro- fero, Sardanaphalus, SmackBot, RDBury, Benjaminb, Incnis Mrsi, Bggoldie~enwiki, Gilliam, Anwar saadat, TimBentley, Octahedron80, Baronnet, Darth Panda, JustUser, Demoeconomist, Cybercobra, SashatoBot, Vriullop, JoshuaZ, Muadd, Laurens-af, Newone, Vanisaac, CmdrObot, MarsRover, TheTito, Gregbard, Grandexandi, Doctormatt, Thijs!bot, Futurebird, Porqin, Seaphoto, John.d.page, Salgueiro~enwiki, JAnDbot, Hut 8.5, Moralist, Magioladitis, VoABot II, Twsx, R'n'B, Pbroks13, J.delanoy, Kimse, SHAN3, Maurice Carbonaro, Drahgo, Bobianite, Barneca, Philip Trueman, TXiKiBoT, Olly150, Siddthekidd, Falcon8765, Symane, Dogah, SieBot, Tiddly Tom, ToePeu.bot, Flyer22, Martarius, Dom96, ClueBot, WDavis1911, Liempt, Alexbot, IJA, Addbot, Some jerk on the Internet, Fgnievinski, Laaknor- Bot, Numbo3-bot, Romanskolduns, Angrysockhop, Math Champion, Luckas-bot, Yobot, Fraggle81, TaBOT-zerem, Reindra, Eric- Wester, AnomieBOT, Rubinbot, Galoubet, Gowr, ArthurBot, Xqbot, Aiuw, RibotBOT, Brayan Jaimes, Aaron Kauppi, MacMan4891, BoomerAB, Pinethicket, Piandcompany, Distortiondude, EmausBot, JSquish, AmigoDoPaulo, ZéroBot, D.Lazard, Card Zero, Paul- miko, Chewings72, ClueBot NG, Wcherowi, SusikMkr, Braincricket, Usageunit, Rm1271, David.karpay, Pbierre, Sriharsh1234, Brirush, 54 CHAPTER 5. POINT (GEOMETRY)

Bg9989, Wingconeel, Muhamad ittal, Akwillis and Anonymous: 123

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