Cross sections of a surface z = f(x, y) are obtained by slicing the surface with vertical planes at fixed values of one variable. For a fixed y = c, the cross section is the 2D curve z = f(x, c), which is a function of x alone.
This visualization shows the surface f(x, y) = x^2 + y^2 (a circular paraboloid) together with its cross sections at various fixed y-values, helping students understand how a 3D surface can be decomposed into a family of 2D curves.
For the paraboloid, each cross section at y = c is a parabola z = x^2 + c^2, shifted upward by c^2.
The transparent khaki surface shows the paraboloid, with colored curves drawn at y = -4, -3, -2, ..., 3. Each curve is a parabolic cross section of the surface.
Each panel shows a single cross section z = f(x, c) at a fixed y-value. Notice that:
- All cross sections are parabolas (z = x^2 + c^2)
- Cross sections at larger |y| values are shifted upward
- The shape of each cross section is identical -- only the vertical offset changes
| File | Description |
|---|---|
sage_code |
Original SageMath implementation |
plot3d_cross_sections.py |
Python 3 translation with matplotlib |
plot3d_cross_sections.png |
3D surface with cross section curves |
plot3d_cross_sections_slices.png |
Individual 2D cross section panels |
python3 plot3d_cross_sections.py- Cross sections at fixed y are obtained by slicing the surface with vertical planes parallel to the xz-plane
- For symmetric surfaces like x^2 + y^2, cross sections at y = c and y = -c are identical
- Understanding cross sections is essential for computing volumes by slicing in integral calculus
- Cross sections and level curves are complementary: level curves slice horizontally, cross sections slice vertically