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Conic Sections (2D) and Quadric Surfaces

What you will learn today

Conic Sections (in 2D coordinates)

Cylinders (3D)

Quadric Surfaces (3D)

Vectors and the Geometry of Space 1/24 Conic Sections (2D) Cylinders and Quadric Surfaces Shifted Conics

Conic sections result from intersecting a with a .

Vectors and the Geometry of Space 2/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

Vectors and the Geometry of Space 3/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics A is the set of points in a plane that are equidistant from a fixed point F (called the focus) and a fixed line (called the directrix). The point halfway between the focus and the directrix is on the parabola, it is called the vertex. The line perpendicular to the directrix and through the focus is called the axis.

Vectors and the Geometry of Space 4/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

Choose the origin O to be at the vertex, the y-axis as the axis of the parabola, the focus F(0,p). For a point P(x,y) on the parabola, have q |PF | = x2 + (y − p)2 = |y + p| ⇒ x2 = 4py

In general, it opens upward if p¿0 and downward if p¡0. It is symmetric about the y-axis. If we interchange x and y in the equations, y 2 = 4px is the parabola with focus (p,0) and directrix x = −p. It opens to the right if p¿0 and opens to the left if p¡0.

Vectors and the Geometry of Space 5/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

Find the focus and directrix of the parabola y 2 + 10x = 0 and sketch the graph. Reflection properties of parabola.

Vectors and the Geometry of Space 6/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics An is the set of points in a plane the sum of whose distances from two fixed points F1 and F2 is a constant. F1 and F2 are called foci. Kepler’s law 1 says that the orbits of the planets in the solar system are ellipses with the sun at one focus. Put the foci on the x-axis (−c, 0) and (c, 0), the sum of distance is 2a > 0. Therefore the vertices of ellipse are (−a, 0) and (a, 0) on the x-axis. The line segment between the two vertices is called the major axis. When c=0, the two foci coincide and the ellipse is a circle. P(x, y) is on the ellipse,

|PF1| + |PF2| = 2a Put b2 = a2 − c2, the above equation can be written as x2 y 2 + = 1 a2 b2

Vectors and the Geometry of Space 7/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

On the other hand, the ellipse

x2 y 2 + = 1, a ≥ b > 0 b2 a2 has foci (0, ±c) and vertices (0, ±a) on the y-axis.

Vectors and the Geometry of Space 8/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

1. Sketch the graph of 9x2 + 16y 2 = 144 and locate the foci. 2. Find an equation of the ellipse with foci (0, ±2) and vertices (0, ±3). 3. Reflection properties of ellipse.

Vectors and the Geometry of Space 9/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

A is the set of all points in a plane the difference of whose distances from two fixed points F1 and F2 is a constant.

Vectors and the Geometry of Space 10/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

Put the vertices F1, F2 = (±c, 0) along the x-axis, the vertices are (±a, 0). Have

|PF1| − |PF2| = ±2a Write b2 = c2 − a2. The equation of the parabola is

x2 y 2 − = 1 a2 b2 A hyperbola has two branches. It has asymptotes b y = ± x a

Vectors and the Geometry of Space 11/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

On the other hand, the hyperbola with foci (0, ±c), vertices (0, ±a) on the y-axis has form

y 2 x2 − = 1 a2 b2 a It has asymptotes y = ± b x

Vectors and the Geometry of Space 12/24 Parabolas Conic Sections (2D) ellipses Cylinders and Quadric Surfaces Hyperbolas Shifted Conics

If we shift the origin h units to the right and k units up, we will replace x and y by x-h and y-k in the equations. Sketch the conic 9x2 − 4y 2 − 72x + 8y + 176 = 0

Vectors and the Geometry of Space 13/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

General surface: To Sketch the graph of a surface, it is useful to determine the curves of intersection if the surface with planes parallel to the coordinate planes. These curves are called traces (or cross-sections) of the surface. Cylinders: A is a surface that consists all lines (called rulings) that are parallel to a given line and pass through a given . The parabolic cylinder: x = ay 2, a > 0

Vectors and the Geometry of Space 14/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

When one of the variable x, y or z (say x) is missing from the equation, then the surface is a cylinder, and the rulings are parallel to (say x). Example: Sketch x2 + y 2 = 1.

Vectors and the Geometry of Space 15/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

A quadric surface is the graph of a second-degree equation in three variables x, y and z. The most general such equation is

Ax2 + By 2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0 where the capital letters are constants (some of them could be 0!). By translation and rotation it can be brought into the standard forms Ax2 + By 2 + Cz2 + J = 0 or Ax2 + By 2 + Iz = 0

Vectors and the Geometry of Space 16/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Example: use traces to sketch the surface with equation x2 y 2 z2 + + = 1 4 9 4

In general, an has equation x2 y 2 z2 + + = 1 a2 b2 c2 Vectors and the Geometry of Space 17/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Example: z = 4x2 + y 2 This is called an elliptic .

Vectors and the Geometry of Space 18/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Example: z = y 2 − x2 This is called a hyperbolic paraboloid (or saddle),

Vectors and the Geometry of Space 19/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Example: x2 z2 + y 2 − = 1 4 4 of one sheet,

Vectors and the Geometry of Space 20/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Example: 4x2 − y 2 + 2z2 + 4 = 0 Hyperboloid of two sheets,

whether a hyperboloid is one sheet or two sheets depends on whether z could be 0 or not.

Vectors and the Geometry of Space 21/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Example: : x2 y 2 z2 + − = 0 a2 b2 c2

Vectors and the Geometry of Space 22/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

Applications: A satellite dish is a paraboloid. Cooling towers in power stations is in the shape of . Twin gears in a triturating juicer,

Vectors and the Geometry of Space 23/24 Conic Sections (2D) Quadric Surfaces Cylinders and Quadric Surfaces

What you have learned today

Conic Sections (in 2D coordinates)

Cylinders (3D)

Quadric Surfaces (3D)

Vectors and the Geometry of Space 24/24