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Chapter 11. Three Dimensional Analytic Geometry and Vectors

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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. + − =1 a2 b2 c2 x2 y2 z2 2. − + =1 a2 b2 c2 x2 y2 z2 3. − + + =1 a2 b2 c2 are called hyperbolids of one sheet since two of its traces are hyperbolas and one is an ellipse, and its surface consists of just one piece. 4 2 ±10 ±10 z ±5 ±5 5 5 y x 10 10 ±2 ±4 Example 2. Find the traces of the surface x2 − y2 + z2 =1 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. 2 Hyperboloid of two sheets. The quadric surface with equations x2 y2 z2 1. − − + =1 a2 b2 c2 x2 y2 z2 2. − + − =1 a2 b2 c2 x2 y2 z2 3. − − =1 a2 b2 c2 are called hyperbolids of two sheets since two of its traces are hyperbolas and one is an ellipse, and its surface is two sheets. 10 5 y 5 10 z ±10 ±5 5 10 x ±5 ±10 Example 3. Find the traces of the surface 9x2 − y2 − z2 =9 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. Cone. The quadric surface with equations z2 x2 y2 1. = + c2 a2 b2 y2 x2 z2 2. = + b2 a2 c2 3 x2 y2 z2 3. = + a2 b2 c2 are called cones. 4 x 2 10 5 y ±5 ±10 5 10 ±2 ±4 If P is any point on the cone, then the line OP lies entirely on the cone. The traces in horizontal planes z = k are ellipses and traces in vertical planes x = k or y = k are hyperbolas if k =6 0 but are pairs of lines if k = 0. Paraboloids. The quadric surface with equations z x2 y2 1. = + c a2 b2 y x2 z2 2. = + b a2 c2 x y2 z2 3. = + a b2 c2 are called elliptic paraboloids because its traces in horizontal planes z = k are ellipses, whereas its in vertical planes x = k or y = k are parabolas. 10 8 6 4 2 20 ±10 ±5 5 x y 10 ±20 Example 4. Find the traces of the surface y = x2 + z2 in the planes x = k, y = k, and z = k. Identify the surface and sketch it. 4 The hyperbolic paraboloid z x2 y2 1. = − c a2 b2 y x2 z2 2. = − b a2 c2 x y2 z2 3. = − a b2 c2 also has parabolas as its vertical traces, but its has hyperbolas as its horizontal traces. ±100 ±50 ±10 ±5 ±10 10 5 x y 10 50 100 Quadric cylinders. When one of the variables is missing from the equation of a surface, x2 y2 then the surface is a cylinder. The equation + = 1 represents the elliptic cylinder, the a2 b2 x2 y2 equation − = 1 represents a hyperbolic cylinder, and the equation y = ax2 represents a2 b2 the hyperbolic cylinder. 4 4 4 2 4 2 2 2 ±4 y ±2 z ±5 ±2 ±3 z z ±2 ±1 ±2 ±4 1 2 x 2 2 3 4 ±4 2 y y x 4 x 5 ±2 ±2 ±2 ±4 ±4 ±4 Example 5. Classify the quadric surface 4x2 − y2 + z2 +8x +8z + 24 = 0 and sketch it. 5.
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