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VECTORS AND THE GEOMETRY OF SPACE

VECTORS AND THE GEOMETRY OF SPACE. Cylinders and Quadratic Surfaces. Quadric Surfaces. A quadric surface is the graph of a second degree equation in three variables x , y , and z . The most general equation is: Ax 2 + By 2 + Cz 2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0

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VECTORS AND THE GEOMETRY OF SPACE

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  1. VECTORS AND THE GEOMETRY OF SPACE Cylinders and Quadratic Surfaces

  2. Quadric Surfaces A quadric surface is the graph of a second degree equation in three variables x, y, and z. The most general equation is: Ax2 + By2 + Cz2 + Dxy + Eyz + Fxz + Gx + Hy + Iz + J = 0 where A, B, C, … J are constants.

  3. 25 20 -4 -4 -2 -2 0 2 2 4 4 Paraboloid z = x2 + y2

  4. 10 8 6 4 -4 2 -2 0 -2 2 -4 4 -6 Elliptic Paraboloid z = ax2 + y2 (or z = x2 + ay2)

  5. Parabolic cylinder z = x2

  6. -3 -3 -2 -2 -1 -1 0 1 1 2 2 3 3 Cylinders x2+ y2 = 1 or y2 + z2 = 1 z y x

  7. 1 0 -1 -2 1 -1 0 0 -1 1 2 Ellipsoids If a = b = c, then it would become a sphere.

  8. 5 4 -4 -4 -2 -2 0 2 2 4 4 Cones A circular cone opening upwards where a is a constant.

  9. -15 -10 -5 -10 -5 0 5 10 5 10 15 An elliptic cone opening upwards where a, b are constants

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