Analytic Geometry: the Conic Sections
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Chapter 11. Three Dimensional Analytic Geometry and Vectors
Chapter 11. Three dimensional analytic geometry and vectors. Section 11.5 Quadric surfaces. Curves in R2 : x2 y2 ellipse + =1 a2 b2 x2 y2 hyperbola − =1 a2 b2 parabola y = ax2 or x = by2 A quadric surface is the graph of a second degree equation in three variables. The most general such equation is Ax2 + By2 + Cz2 + Dxy + Exz + F yz + Gx + Hy + Iz + J =0, where A, B, C, ..., J are constants. By translation and rotation the equation can be brought into one of two standard forms Ax2 + By2 + Cz2 + J =0 or Ax2 + By2 + Iz =0 In order to sketch the graph of a quadric surface, it is useful to determine the curves of intersection of the surface with planes parallel to the coordinate planes. These curves are called traces of the surface. Ellipsoids The quadric surface with equation x2 y2 z2 + + =1 a2 b2 c2 is called an ellipsoid because all of its traces are ellipses. 2 1 x y 3 2 1 z ±1 ±2 ±3 ±1 ±2 The six intercepts of the ellipsoid are (±a, 0, 0), (0, ±b, 0), and (0, 0, ±c) and the ellipsoid lies in the box |x| ≤ a, |y| ≤ b, |z| ≤ c Since the ellipsoid involves only even powers of x, y, and z, the ellipsoid is symmetric with respect to each coordinate plane. Example 1. Find the traces of the surface 4x2 +9y2 + 36z2 = 36 1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Hyperboloids Hyperboloid of one sheet. The quadric surface with equations x2 y2 z2 1. -
A Quick Algebra Review
A Quick Algebra Review 1. Simplifying Expressions 2. Solving Equations 3. Problem Solving 4. Inequalities 5. Absolute Values 6. Linear Equations 7. Systems of Equations 8. Laws of Exponents 9. Quadratics 10. Rationals 11. Radicals Simplifying Expressions An expression is a mathematical “phrase.” Expressions contain numbers and variables, but not an equal sign. An equation has an “equal” sign. For example: Expression: Equation: 5 + 3 5 + 3 = 8 x + 3 x + 3 = 8 (x + 4)(x – 2) (x + 4)(x – 2) = 10 x² + 5x + 6 x² + 5x + 6 = 0 x – 8 x – 8 > 3 When we simplify an expression, we work until there are as few terms as possible. This process makes the expression easier to use, (that’s why it’s called “simplify”). The first thing we want to do when simplifying an expression is to combine like terms. For example: There are many terms to look at! Let’s start with x². There Simplify: are no other terms with x² in them, so we move on. 10x x² + 10x – 6 – 5x + 4 and 5x are like terms, so we add their coefficients = x² + 5x – 6 + 4 together. 10 + (-5) = 5, so we write 5x. -6 and 4 are also = x² + 5x – 2 like terms, so we can combine them to get -2. Isn’t the simplified expression much nicer? Now you try: x² + 5x + 3x² + x³ - 5 + 3 [You should get x³ + 4x² + 5x – 2] Order of Operations PEMDAS – Please Excuse My Dear Aunt Sally, remember that from Algebra class? It tells the order in which we can complete operations when solving an equation. -
Lesson 3: Rectangles Inscribed in Circles
NYS COMMON CORE MATHEMATICS CURRICULUM Lesson 3 M5 GEOMETRY Lesson 3: Rectangles Inscribed in Circles Student Outcomes . Inscribe a rectangle in a circle. Understand the symmetries of inscribed rectangles across a diameter. Lesson Notes Have students use a compass and straightedge to locate the center of the circle provided. If necessary, remind students of their work in Module 1 on constructing a perpendicular to a segment and of their work in Lesson 1 in this module on Thales’ theorem. Standards addressed with this lesson are G-C.A.2 and G-C.A.3. Students should be made aware that figures are not drawn to scale. Classwork Scaffolding: Opening Exercise (9 minutes) Display steps to construct a perpendicular line at a point. Students follow the steps provided and use a compass and straightedge to find the center of a circle. This exercise reminds students about constructions previously . Draw a segment through the studied that are needed in this lesson and later in this module. point, and, using a compass, mark a point equidistant on Opening Exercise each side of the point. Using only a compass and straightedge, find the location of the center of the circle below. Label the endpoints of the Follow the steps provided. segment 퐴 and 퐵. Draw chord 푨푩̅̅̅̅. Draw circle 퐴 with center 퐴 . Construct a chord perpendicular to 푨푩̅̅̅̅ at and radius ̅퐴퐵̅̅̅. endpoint 푩. Draw circle 퐵 with center 퐵 . Mark the point of intersection of the perpendicular chord and the circle as point and radius ̅퐵퐴̅̅̅. 푪. Label the points of intersection . -
Squaring the Circle a Case Study in the History of Mathematics the Problem
Squaring the Circle A Case Study in the History of Mathematics The Problem Using only a compass and straightedge, construct for any given circle, a square with the same area as the circle. The general problem of constructing a square with the same area as a given figure is known as the Quadrature of that figure. So, we seek a quadrature of the circle. The Answer It has been known since 1822 that the quadrature of a circle with straightedge and compass is impossible. Notes: First of all we are not saying that a square of equal area does not exist. If the circle has area A, then a square with side √A clearly has the same area. Secondly, we are not saying that a quadrature of a circle is impossible, since it is possible, but not under the restriction of using only a straightedge and compass. Precursors It has been written, in many places, that the quadrature problem appears in one of the earliest extant mathematical sources, the Rhind Papyrus (~ 1650 B.C.). This is not really an accurate statement. If one means by the “quadrature of the circle” simply a quadrature by any means, then one is just asking for the determination of the area of a circle. This problem does appear in the Rhind Papyrus, but I consider it as just a precursor to the construction problem we are examining. The Rhind Papyrus The papyrus was found in Thebes (Luxor) in the ruins of a small building near the Ramesseum.1 It was purchased in 1858 in Egypt by the Scottish Egyptologist A. -
Automatic Estimation of Sphere Centers from Images of Calibrated Cameras
Automatic Estimation of Sphere Centers from Images of Calibrated Cameras Levente Hajder 1 , Tekla Toth´ 1 and Zoltan´ Pusztai 1 1Department of Algorithms and Their Applications, Eotv¨ os¨ Lorand´ University, Pazm´ any´ Peter´ stny. 1/C, Budapest, Hungary, H-1117 fhajder,tekla.toth,[email protected] Keywords: Ellipse Detection, Spatial Estimation, Calibrated Camera, 3D Computer Vision Abstract: Calibration of devices with different modalities is a key problem in robotic vision. Regular spatial objects, such as planes, are frequently used for this task. This paper deals with the automatic detection of ellipses in camera images, as well as to estimate the 3D position of the spheres corresponding to the detected 2D ellipses. We propose two novel methods to (i) detect an ellipse in camera images and (ii) estimate the spatial location of the corresponding sphere if its size is known. The algorithms are tested both quantitatively and qualitatively. They are applied for calibrating the sensor system of autonomous cars equipped with digital cameras, depth sensors and LiDAR devices. 1 INTRODUCTION putation and memory costs. Probabilistic Hough Transform (PHT) is a variant of the classical HT: it Ellipse fitting in images has been a long researched randomly selects a small subset of the edge points problem in computer vision for many decades (Prof- which is used as input for HT (Kiryati et al., 1991). fitt, 1982). Ellipses can be used for camera calibra- The 5D parameter space can be divided into two tion (Ji and Hu, 2001; Heikkila, 2000), estimating the pieces. First, the ellipse center is estimated, then the position of parts in an assembly system (Shin et al., remaining three parameters are found in the second 2011) or for defect detection in printed circuit boards stage (Yuen et al., 1989; Tsuji and Matsumoto, 1978). -
Exploding the Ellipse Arnold Good
Exploding the Ellipse Arnold Good Mathematics Teacher, March 1999, Volume 92, Number 3, pp. 186–188 Mathematics Teacher is a publication of the National Council of Teachers of Mathematics (NCTM). More than 200 books, videos, software, posters, and research reports are available through NCTM’S publication program. Individual members receive a 20% reduction off the list price. For more information on membership in the NCTM, please call or write: NCTM Headquarters Office 1906 Association Drive Reston, Virginia 20191-9988 Phone: (703) 620-9840 Fax: (703) 476-2970 Internet: http://www.nctm.org E-mail: [email protected] Article reprinted with permission from Mathematics Teacher, copyright May 1991 by the National Council of Teachers of Mathematics. All rights reserved. Arnold Good, Framingham State College, Framingham, MA 01701, is experimenting with a new approach to teaching second-year calculus that stresses sequences and series over integration techniques. eaders are advised to proceed with caution. Those with a weak heart may wish to consult a physician first. What we are about to do is explode an ellipse. This Rrisky business is not often undertaken by the professional mathematician, whose polytechnic endeavors are usually limited to encounters with administrators. Ellipses of the standard form of x2 y2 1 5 1, a2 b2 where a > b, are not suitable for exploding because they just move out of view as they explode. Hence, before the ellipse explodes, we must secure it in the neighborhood of the origin by translating the left vertex to the origin and anchoring the left focus to a point on the x-axis. -
Writing the Equation of a Line
Name______________________________ Writing the Equation of a Line When you find the equation of a line it will be because you are trying to draw scientific information from it. In math, you write equations like y = 5x + 2 This equation is useless to us. You will never graph y vs. x. You will be graphing actual data like velocity vs. time. Your equation should therefore be written as v = (5 m/s2) t + 2 m/s. See the difference? You need to use proper variables and units in order to compare it to theory or make actual conclusions about physical principles. The second equation tells me that when the data collection began (t = 0), the velocity of the object was 2 m/s. It also tells me that the velocity was changing at 5 m/s every second (m/s/s = m/s2). Let’s practice this a little, shall we? Force vs. mass F (N) y = 6.4x + 0.3 m (kg) You’ve just done a lab to see how much force was necessary to get a mass moving along a rough surface. Excel spat out the graph above. You labeled each axis with the variable and units (well done!). You titled the graph starting with the variable on the y-axis (nice job!). Now we turn to the equation. First we replace x and y with the variables we actually graphed: F = 6.4m + 0.3 Then we add units to our slope and intercept. The slope is found by dividing the rise by the run so the units will be the units from the y-axis divided by the units from the x-axis. -
Algebraic Study of the Apollonius Circle of Three Ellipses
EWCG 2005, Eindhoven, March 9–11, 2005 Algebraic Study of the Apollonius Circle of Three Ellipses Ioannis Z. Emiris∗ George M. Tzoumas∗ Abstract methods readily extend to arbitrary inputs. The algorithms for the Apollonius diagram of el- We study the external tritangent Apollonius (or lipses typically use the following 2 main predicates. Voronoi) circle to three ellipses. This problem arises Further predicates are examined in [4]. when one wishes to compute the Apollonius (or (1) given two ellipses and a point outside of both, Voronoi) diagram of a set of ellipses, but is also of decide which is the ellipse closest to the point, independent interest in enumerative geometry. This under the Euclidean metric paper is restricted to non-intersecting ellipses, but the extension to arbitrary ellipses is possible. (2) given 4 ellipses, decide the relative position of the We propose an efficient representation of the dis- fourth one with respect to the external tritangent tance between a point and an ellipse by considering a Apollonius circle of the first three parametric circle tangent to an ellipse. The distance For predicate (1) we consider a circle, centered at the of its center to the ellipse is expressed by requiring point, with unknown radius, which corresponds to the that their characteristic polynomial have at least one distance to be compared. A tangency point between multiple real root. We study the complexity of the the circle and the ellipse exists iff the discriminant tritangent Apollonius circle problem, using the above of the corresponding pencil’s determinant vanishes. representation for the distance, as well as sparse (or Hence we arrive at a method using algebraic numbers toric) elimination. -
The Geometry of the Phase Diffusion Equation
J. Nonlinear Sci. Vol. 10: pp. 223–274 (2000) DOI: 10.1007/s003329910010 © 2000 Springer-Verlag New York Inc. The Geometry of the Phase Diffusion Equation N. M. Ercolani,1 R. Indik,1 A. C. Newell,1,2 and T. Passot3 1 Department of Mathematics, University of Arizona, Tucson, AZ 85719, USA 2 Mathematical Institute, University of Warwick, Coventry CV4 7AL, UK 3 CNRS UMR 6529, Observatoire de la Cˆote d’Azur, 06304 Nice Cedex 4, France Received on October 30, 1998; final revision received July 6, 1999 Communicated by Robert Kohn E E Summary. The Cross-Newell phase diffusion equation, (|k|)2T =∇(B(|k|) kE), kE =∇2, and its regularization describes natural patterns and defects far from onset in large aspect ratio systems with rotational symmetry. In this paper we construct explicit solutions of the unregularized equation and suggest candidates for its weak solutions. We confirm these ideas by examining a fourth-order regularized equation in the limit of infinite aspect ratio. The stationary solutions of this equation include the minimizers of a free energy, and we show these minimizers are remarkably well-approximated by a second-order “self-dual” equation. Moreover, the self-dual solutions give upper bounds for the free energy which imply the existence of weak limits for the asymptotic minimizers. In certain cases, some recent results of Jin and Kohn [28] combined with these upper bounds enable us to demonstrate that the energy of the asymptotic minimizers converges to that of the self-dual solutions in a viscosity limit. 1. Introduction The mathematical models discussed in this paper are motivated by physical systems, far from equilibrium, which spontaneously form patterns. -
Device Constructions with Hyperbolas
Device Constructions with Hyperbolas Alfonso Croeze1 William Kelly1 William Smith2 1Department of Mathematics Louisiana State University Baton Rouge, LA 2Department of Mathematics University of Mississippi Oxford, MS July 8, 2011 Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas Hyperbola Definition Conic Section Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas Hyperbola Definition Conic Section Two Foci Focus and Directrix Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas The Project Basic constructions Constructing a Hyperbola Advanced constructions Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas Rusty Compass Theorem Given a circle centered at a point A with radius r and any point C different from A, it is possible to construct a circle centered at C that is congruent to the circle centered at A with a compass and straightedge. Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas X B C A Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas X D B C A Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas A B X Y Angle Duplication A X Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas Angle Duplication A X A B X Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas C Z A B X Y C Z A B X Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas C Z A B X Y C Z A B X Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas Constructing a Perpendicular C C A B Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas C C X X O A B A B Y Y Croeze, Kelly, Smith LSU&UoM Device Constructions with Hyperbolas We needed a way to draw a hyperbola. -
Chapter 10 Analytic Geometry
Chapter 10 Analytic Geometry Section 10.1 13. (e); the graph has vertex ()()hk,1,1= and opens to the right. Therefore, the equation of the graph Not applicable has the form ()yax−=142 () − 1. Section 10.2 14. (d); the graph has vertex ()()hk,0,0= and ()()−+−22 1. xx21 yy 21 opens down. Therefore, the equation of the graph has the form xay2 =−4 . The graph passes 2 −4 through the point ()−−2, 1 so we have 2. = 4 2 2 ()−=−−241a () 2 = 3. ()x +=49 44a = x +=±43 a 1 2 xx+=43 or +=− 4 3 Thus, the equation of the graph is xy=−4 . xx=−1 or =− 7 15. (h); the graph has vertex ()(hk,1,1=− − ) and The solution set is {7,1}−−. opens down. Therefore, the equation of the graph 2 4. (2,5)−− has the form ()xay+=−+141 (). 5. 3, up 16. (a); the graph has vertex ()()hk,0,0= and 6. parabola opens to the right. Therefore, the equation of the graph has the form yax2 = 4 . The graph passes 7. c through the point ()1, 2 so we have 8. (3, 2) ()2412 = a () 44= a 9. (3,6) 1 = a 2 10. y =−2 Thus, the equation of the graph is yx= 4 . 11. (b); the graph has a vertex ()()hk,0,0= and 17. (c); the graph has vertex ()()hk,0,0= and opens up. Therefore, the equation of the graph opens to the left. Therefore, the equation of the 2 has the form xay2 = 4 . The graph passes graph has the form yax=−4 . -
Calculus Terminology
AP Calculus BC Calculus Terminology Absolute Convergence Asymptote Continued Sum Absolute Maximum Average Rate of Change Continuous Function Absolute Minimum Average Value of a Function Continuously Differentiable Function Absolutely Convergent Axis of Rotation Converge Acceleration Boundary Value Problem Converge Absolutely Alternating Series Bounded Function Converge Conditionally Alternating Series Remainder Bounded Sequence Convergence Tests Alternating Series Test Bounds of Integration Convergent Sequence Analytic Methods Calculus Convergent Series Annulus Cartesian Form Critical Number Antiderivative of a Function Cavalieri’s Principle Critical Point Approximation by Differentials Center of Mass Formula Critical Value Arc Length of a Curve Centroid Curly d Area below a Curve Chain Rule Curve Area between Curves Comparison Test Curve Sketching Area of an Ellipse Concave Cusp Area of a Parabolic Segment Concave Down Cylindrical Shell Method Area under a Curve Concave Up Decreasing Function Area Using Parametric Equations Conditional Convergence Definite Integral Area Using Polar Coordinates Constant Term Definite Integral Rules Degenerate Divergent Series Function Operations Del Operator e Fundamental Theorem of Calculus Deleted Neighborhood Ellipsoid GLB Derivative End Behavior Global Maximum Derivative of a Power Series Essential Discontinuity Global Minimum Derivative Rules Explicit Differentiation Golden Spiral Difference Quotient Explicit Function Graphic Methods Differentiable Exponential Decay Greatest Lower Bound Differential