Quadric Surfaces Mind Map
Algorithm & Systems VisualizationsEvery quadric surfaces visualization on this site on one map. The branches are the main ideas. Each leaf is a page; the grey line under it is the one thing to remember about it. Click a branch to focus on it, click a page to open it.
1. How the map is organized
Write the surface in standard form and ask two questions:
- How many variables are squared? All three: three squares. Two squared and one to the first power: paraboloids. One variable missing: cylinders.
- What are the signs? All plus is an ellipsoid, one minus one sheet, two minus two sheets, and = 0 a cone; for paraboloids, same signs a bowl, opposite a saddle.
- If the equation has cross terms like xy, rotate first: that is build and classify. Degree 3 or 4 is higher degree.
2. The branches side by side
| Branch | Standard form | Cut z = h | Example |
|---|---|---|---|
| Start here | Ax + By + Cz + D = 0 | a line | plane |
| Three squares | ±x²/a² ± y²/b² ± z²/c² = 1 or 0 | ellipse, point or nothing | ellipsoid |
| Paraboloids | z = ±x²/a² ± y²/b² | ellipse or hyperbola | saddle |
| Cylinders | x²/a² ± y²/b² = 1 | the same curve every h | tube |
| Build and classify | general form with xy, yz, zx | depends on the type | rotated quadric |
| Higher degree | F(x, y, z) = 0, degree 3 or 4 | cubic or quartic curve | Kummer surface |
3. Five facts to remember
- A quadric is the set where a degree-2 polynomial in x, y, z is zero.
- Every cut of a quadric by a plane is a conic: ellipse, parabola, hyperbola or a degenerate one.
- The hyperboloid of one sheet and the saddle are doubly ruled: two straight lines pass through every point.
- The eigenvalues of the symmetric matrix of the x², xy … terms give the principal axes and the type.
- A straight line meets a surface of degree n in at most n points, unless it lies on it.
See all quadric surfaces pages in the index, or the other mind maps.