What you will learn today Conic Sections (in 2D coordinates) Cylinders (3D) Quadric Surfaces (3D) Vectors and the Geometry of Space 1/24
Parabolas ellipses Hyperbolas Shifted Conics Conic sections result from intersecting a cone with a plane. Vectors and the Geometry of Space 2/24
Parabolas ellipses Hyperbolas Shifted Conics Vectors and the Geometry of Space 3/24
Parabolas ellipses Hyperbolas Shifted Conics A parabola 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 ellipses 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 PF = x 2 + (y p) 2 = y + p x 2 = 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 ellipses 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 ellipses Hyperbolas Shifted Conics An ellipse is the set of points in a plane the sum of whose distances from two fixed points F 1 and F 2 is a constant. F 1 and F 2 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, PF 1 + PF 2 = 2a Put b 2 = a 2 c 2, the above equation can be written as x 2 a 2 + y 2 b 2 = 1 Vectors and the Geometry of Space 7/24
Parabolas ellipses Hyperbolas Shifted Conics On the other hand, the ellipse x 2 b 2 + y 2 a 2 = 1, a b > 0 has foci (0, ±c) and vertices (0, ±a) on the y-axis. Vectors and the Geometry of Space 8/24
Parabolas ellipses Hyperbolas Shifted Conics 1. Sketch the graph of 9x 2 + 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 ellipses Hyperbolas Shifted Conics A hyperbola is the set of all points in a plane the difference of whose distances from two fixed points F 1 and F 2 is a constant. Vectors and the Geometry of Space 10/24
Parabolas ellipses Hyperbolas Shifted Conics Put the vertices F 1, F 2 = (±c, 0) along the x-axis, the vertices are (±a, 0). Have PF 1 PF 2 = ±2a Write b 2 = c 2 a 2. The equation of the parabola is x 2 a 2 y 2 b 2 = 1 A hyperbola has two branches. It has asymptotes y = ± b a x Vectors and the Geometry of Space 11/24
Parabolas ellipses Hyperbolas Shifted Conics On the other hand, the hyperbola with foci (0, ±c), vertices (0, ±a) on the y-axis has form It has asymptotes y = ± a b x y 2 a 2 x 2 b 2 = 1 Vectors and the Geometry of Space 12/24
Parabolas ellipses 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 9x 2 4y 2 72x + 8y + 176 = 0 Vectors and the Geometry of Space 13/24
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 cylinder is a surface that consists all lines (called rulings) that are parallel to a given line and pass through a given plane curve. The parabolic cylinder: x = ay 2, a > 0 Vectors and the Geometry of Space 14/24
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 x 2 + y 2 = 1. Vectors and the Geometry of Space 15/24
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 Ax 2 + By 2 + Cz 2 + 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 Ax 2 + By 2 + Cz 2 + J = 0 or Ax 2 + By 2 + Iz = 0 Vectors and the Geometry of Space 16/24
Quadric Surfaces Example: use traces to sketch the surface with equation x 2 4 + y 2 9 + z2 4 = 1 In general, an ellipsoid has equation x 2 a 2 + y 2 b 2 + z2 c 2 = 1 Vectors and the Geometry of Space 17/24
Quadric Surfaces Example: z = 4x 2 + y 2 This is called an elliptic paraboloid. Vectors and the Geometry of Space 18/24
Quadric Surfaces Example: z = y 2 x 2 This is called a hyperbolic paraboloid (or saddle), Vectors and the Geometry of Space 19/24
Quadric Surfaces Example: x 2 4 + y 2 z2 4 = 1 Hyperboloid of one sheet, Vectors and the Geometry of Space 20/24
Quadric Surfaces Example: Hyperboloid of two sheets, 4x 2 y 2 + 2z 2 + 4 = 0 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
Quadric Surfaces Example: Cones: x 2 a 2 + y 2 b 2 z2 c 2 = 0 Vectors and the Geometry of Space 22/24
Quadric Surfaces Applications: A satellite dish is a paraboloid. Cooling towers in power stations is in the shape of hyperboloids. Twin gears in a triturating juicer, Vectors and the Geometry of Space 23/24
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