A perspective view is good at showing that a surface has shape and bad at telling you exactly what that shape is. The fix is old: slice the surface with a plane and look at the cut. In multivariable calculus this is a trace, in engineering it is a section, and in this calculator it is the Cross-section control in graph settings, which adds a 2D inset below the 3D view.
Three cuts, three questions
Holding x fixed at a value a gives z = f(a, y), a function of y alone. Plotting it in the plane shows how the height changes as you walk parallel to the y-axis at that x. Holding y fixed gives z = f(x, b), the walk parallel to the x-axis. Both are vertical cuts. The third kind, holding z fixed at c, solves f(x, y) = c and gives a level curve in the xy-plane.
| Cut | Curve | Question it answers |
|---|---|---|
| x = a | z = f(a, y) | How does height change along y at this x? |
| y = b | z = f(x, b) | How does height change along x at this y? |
| z = c | f(x, y) = c | Where is the height exactly c? |
The first two are the visual form of partial derivatives. The slope of the curve z = f(a, y) at a point is the partial derivative with respect to y, because x is not changing along that walk. This is why the section panel is the fastest way to check a sign: if the curve is rising at the point, the partial derivative there is positive.
Using the panel
- Open Graph settings and switch Cross-section on. A 2D inset appears under the controls.
- Choose the cutting axis with the X, Y or Z buttons. X and Y cut vertically; Z cuts horizontally.
- Drag the slider. The label shows the exact plane position, such as x = 1.20.
- Read the inset. It draws the cut as a flat curve with its own axes, so you are no longer estimating from perspective.
- Change the axis and compare. The difference between the two vertical cuts is usually the lesson.
The slider bounds are not arbitrary. For an x or y cut they follow the plotted range, so you cannot ask for a plane outside the surface. For a z cut the bounds follow the minimum and maximum heights the current mesh actually reaches, so the plane always has something to cut. A completely flat surface would give a zero-width slider, so the bounds widen slightly in that case.
Worked example: the saddle, cut two ways
Plot z = x^2 - y^2 and cut at x = 0. The inset shows z = -y^2, a downward parabola with its maximum at the origin. Now cut at y = 0. The inset shows z = x^2, an upward parabola with its minimum at the origin. The same surface, the same point, two opposite conclusions about whether the point is high or low.
That pair of pictures is the definition of a saddle point in visual form, and it is more convincing than a rotated 3D view because neither inset depends on camera angle. The algebra agrees: the second derivative with respect to x at the origin is 2, and with respect to y it is -2, so the curvatures have opposite signs.
A horizontal cut is a contour
Set the axis to Z and drag to z = 4 on the same saddle. The inset shows the level curve x^2 - y^2 = 4, a hyperbola opening left and right. Drag to z = 0 and it degenerates into the two lines y = x and y = -x that cross at the origin. Drag to z = -4 and the hyperbola opens the other way. This is exactly what the contour map draws on the floor, seen one level at a time.
Reading the horizontal cut is a different skill from reading the vertical ones. A vertical cut shows height as a curve; a horizontal cut shows where a height occurs as a set of positions. Students who can move between the two have understood the relationship between a surface and its level sets, which is the point of the contour map guide.
Continuity and corners, seen directly
Cuts are also the honest way to look at non-smooth surfaces. Plot z = abs(x) + abs(y) and cut at y = 0: the inset is a V, with a corner at the origin and no single slope there. Cut at x = 0 and you get the same V rotated. The 3D view shows creases, but the inset shows a function that a first-year student already knows how to analyse.
The same applies to a cone. Cutting z = sqrt(x^2 + y^2) at y = 0 gives z = |x|, which is a V rather than a smooth curve, and that corner is the visual statement that the cone has no tangent plane at its tip. A cut at y = 1 gives the hyperbola branch z = sqrt(x^2 + 1), which is smooth everywhere. One surface, smooth in some slices and not in others.
Limits of the tool
- One section at a time: comparing two cuts means switching the axis, not seeing both at once.
- The inset is a 2D curve, not a filled region, so it shows the boundary of the cut rather than the whole cross-section area.
- A section plane outside the plotted range is clamped, which is a guard rather than an error.
- The curve is drawn from the same sample grid as the surface, so a feature narrower than the grid may be missed in the inset too.
None of those limits changes the main value, which is that a perspective judgement becomes a flat one. When a question asks about behaviour along a line, the inset answers it directly, and the 3D view goes back to doing what it is good at: showing how the slices fit together.