quadric surfaces cheat sheet, Cheat Sheet of Mathematics

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Quadric Surface Examples

Example 1 (12.6.30).

4 x^2 − y + 2z^2 = 0 ⇔ y = x

2 1 / 4 +^

z^2 1 / 2 This is a paraboloid with vertex at the origin, opening along the y-axis.

Figure 1: Paraboloid

Example 2 (12.6.36).

x^2 − y^2 + z^2 − 2 x + 2y + 4z + 2 = 0 ⇔ (x^ −^ 1)

2 2 −^

(y − 1)^2 2 +

(z + 2)^2 2 = 1

This is a hyperboloid of one sheet centered at (1, 1 , −2).

Example 3 (12.6.15). −x^2 + 4y^2 − z^2 = 4 ⇔

This is a hyperboloid of two sheets centered at the origin.

Figure 3: Hyperboloid of two sheets

Example 4 (12.rev.35). 4 x^2 + 4y^2 − 8 y + z^2 = 0 ⇔

This is an ellipsoid centered at (0, 1 , 0).

Figure 4: Ellipsoid

Example 5 (12.rev.31). x^2 = y^2 + 4z^2 ⇔

This is a cone with vertex at the origin and axis parallel to the x-axis.

Figure 5: Cone

Example 6 (12.6.20). x = y^2 − z^2 vs. y = z^2 − x^2

This are both hyperbolic paraboloids, one first with the x-axis perpendicular to the ‘saddle’ and the other with the y-axis perpendicular to the ‘saddle’.

(a) x-axis perpendicular to the ‘saddle’ (b) y-axis perpendicular to the ‘saddle’ Figure 6: Hyperbolic Paraboloids