Vector Calc - 3D Surfaces

Paraboloid

2 squares (same sign), # output (ex. 2x² + y² - z = 1)

Hyperboloid of one sheet

3 squares (two positive, one negative), positive # output (ex. 3x² + y² - 2z² = 1)

Hyperboloid of two sheets

3 squares (two positive, one negative), negative # output (ex. x² + (y-2)² - 4z² = -1)

Cone

3 squares (two positive, one negative), 0 output (ex. (x-1)² + 5y² - z² = 0)

Hyperbolic paraboloid (a.k.a saddle)

2 squares (opposite sign), # output (ex. -x² + 3(y+1)² - z = 4)

Ellipsoid (includes sphere)

3 squares (all positive), positive # output (ex. 3x² + 2y² + z² = 1)

Plane

No squares (ex. x + 3y - 5z + 4 = 0)

Cylinder on a curve

One of z, y, or z doesn’t appear (ex. x² + z² = 1)

Paraboloid

Hyperboloid of one sheet

Hyperboloid of two sheets

Cone

Hyperbolic paraboloid (a.k.a saddle)

Ellipsoid (includes sphere)

Plane

Cylinder on a curve

2 squares (same sign), # output (ex. 2x² + y² - z = 1)

3 squares (two positive, one negative), positive # output (ex. 3x² + y² - 2z² = 1)

3 squares (two positive, one negative), negative # output (ex. x² + (y-2)² - 4z² = -1)

3 squares (two positive, one negative), 0 output (ex. (x-1)² + 5y² - z² = 0)

2 squares (opposite sign), # output (ex. -x² + 3(y+1)² - z = 4)

3 squares (all positive), positive # output (ex. 3x² + 2y² + z² = 1)

No squares (ex. x + 3y - 5z + 4 = 0)

One of z, y, or z doesn’t appear (ex. x² + z² = 1)