Extending Surface Equations

Integrated Math 4

Mrs. Tyrpak

Nonlinear Surface Equations

Recall an equation for an ellipse in 2-space:

(This means the major axis = 8 units and the minor axis = 6 units)

Nonlinear Surface Equations

What equation would an ellipsoid have in 3-space with the following measurements:

1. Axis along x-axis = 8

2. Axis along y-axis = 6

3. Axis along z-axis = 4

Symmetry in 3-space

Recall symmetry in 2-space:

Is this ellipse symmetric with respect to the x-axis? The y-axis?

Symmetry in 3-space

Is the ellipsoid symmetric with respect to the xz-plane? the yz-plane? xy-plane?

Matching

We will use our knowledge of intercepts, symmetry, and traces to match with the following surfaces.

Surfaces for Matching

1. Double Cone

2. Hyperboloid of 1 sheet

3. Paraboloid

4. Elliptical Paraboloid

𝑥2+𝑦2−𝑧 2=0Intercepts:

Traces:

Symmetry:

𝑥2+4 𝑦2− 𝑧2=4Intercepts:

Traces:

Symmetry:

𝑥2+2 𝑦2+3 𝑧=6Intercepts:

Traces:

Symmetry:

𝑥2+𝑦2−𝑧=0Intercepts:

Traces:

Symmetry:

Two Ways Plane Curves can generate surfaces:

1. A line (perpendicular to a plane in which the curve is drawn), to trace the curve generating a cylindrical surface

2. Rotate a curve about a line to get a surface of revolution

Examples:

Cylindrical Surfaces:

Surface of Revolution:

You know what I’m going to say!

Awesome job!!

Don’t forget to complete your extension and enrichment worksheets before you

move on.

Remember you are a mathematician