Surface Gallery

Calculus 3 · Axiom Academy

Every surface in 3D is just a stack of curves. Watch quadrics build themselves from their traces — then slice and read them yourself. A surface is a stack of curves you already know In two dimensions you graph a curve. In three, you graph a whole surface — and it can feel impossible to picture from an equation alone. The trick every mathematician uses: don't try to see the whole thing at once. Slice it into flat curves you already recognize, then stack them back up. Watch four classic quadric surfaces build themselves. Each one is drawn as a stack of horizontal rings (its traces at different heights) threaded by vertical ribs , spinning so you can see the full shape. Notice how the equation and the ring pattern go hand in hand. Four surfaces, one idea: read the rings and you've read the shape. Cut it with a plane — out falls a trace Slide the cutting plane up and down through the surface. Wherever it crosses, you get a flat curve called a trace . Watch what shape the trace is, and how its size changes with height — that pattern is the whole personality of the surface. Stack every one of these circles up the axis and you've rebuilt the whole surface. Go the other way: equation to shape Pick a surface and look at its three coordinate-plane traces — what you'd see slicing on the xy , xz , and yz planes. With a little practice you can read the shape straight off the signs of the squared terms, no plotting required.

This is the written version of the interactive lesson above. See the full Calculus 3 course.