The 3D Geometry Board, Explained

After reading this you will be able to slice a cone with a plane, name the conic that falls out, place its two Dandelin spheres by hand, and check that the sum of focal distances stays constant to four decimal places.

What the board does, and one hook

The 3D Geometry Board is a construction surface in space. You place points, join them into lines and planes, grow spheres and polyhedra, then intersect any two objects and measure the result. The camera orbits, so you see the same construction from every angle. Nothing is baked in: drag a point and every object built from it moves too.

Here is the hook. Take a right circular cone and cut it with a slanted plane. The section is an ellipse. Now drop the two spheres that fit snugly inside the cone and just touch the cutting plane, one above the cut and one below. Each sphere kisses the plane at a single point. Those two points are the foci of the ellipse. Put a point on the ellipse, measure its distance to each focus, add them, and the sum never changes as you drag the point around. That sum equals the length of the major axis. The board lets you watch this stay fixed while everything else moves.

When to reach for it, and when not

Use the board when a relationship in space is hard to see on paper: the shape of a cross-section, why three planes meet in a point or a line, how a net folds back into a solid, or why the conic sections are one family. The orbiting orthographic camera is honest about parallel lines and equal lengths, so you can trust what you compare by eye.

Do not use it as a proof engine. The relation checker tests whether collinearity, coplanarity, parallelism, perpendicularity or tangency survive random legal moves, which is strong evidence but not a proof. Do not expect it to handle non-convex solids for cross-sections or nets; the algorithms assume convexity. And the painter's algorithm sorts translucent faces by depth, so two interpenetrating solids can occasionally draw in the wrong order. For flat constructions you are better served by the Dynamic Geometry Board.

The conic sections from one cone

Set the cone with apex at the origin, axis along the vertical, and half-angle \alpha: the angle between the axis and a generator line. Every point on the cone satisfies

Here x, y, z are the coordinates of a point on the surface, and \tan\alpha is the ratio of radius to height along the cone. Now cut with a plane that makes angle \beta with the axis. The type of conic depends only on how \beta compares to \alpha.

Circle

The plane is perpendicular to the axis, so \beta = 90^\circ.

Ellipse

The plane is steeper than the generators: \alpha \lt \beta \lt 90^\circ. The section is closed.

Parabola

The plane is exactly parallel to one generator: \beta = \alpha. The section is open, one branch.

Hyperbola

The plane is shallower than the generators: \beta \lt \alpha. The plane meets both nappes, giving two branches.

The board samples the conic along 240 generators of the cone, so a hyperbola shows both nappes and is clipped where the generators leave the drawn part of the cone.

Why the Dandelin spheres are the foci

The Dandelin construction is the cleanest proof that a slanted section of a cone is an ellipse in the classical sense: a curve where two focal distances sum to a constant. Inscribe two spheres in the cone, each tangent to the cone along a full circle, and each tangent to the cutting plane at one point. Call those tangent points F_1 and F_2.

Pick any point P on the section. Draw the generator line of the cone through P; it runs from the apex down the surface and crosses both tangent circles, say at A on the upper sphere's circle and B on the lower sphere's circle. Two tangent segments from a single external point to one sphere have equal length. So PF_1 = PA and PF_2 = PB. Adding,

The segment AB is the piece of the generator between the two tangent circles. That length is the same for every generator, because the circles are fixed. So the sum PF_1 + PF_2 is constant as P moves. That constant equals the major axis length 2a. The two tangent points are the foci by definition.

A worked example you can reproduce

Reading and interpreting the numbers

The measure tools report distances, angles between lines and planes, areas and volumes. A few sanity checks keep you honest.

Angle between a line and a plane is measured from the plane, not its normal, so a line lying in the plane reads 0 and a line piercing straight through reads 90. Angle between two planes is the dihedral angle along their line of intersection. Two parallel planes give an intersection tool that returns nothing, which is the correct answer, not a bug.

For volumes, check against known formulas. A cone of radius 1 and height 3 has volume \tfrac{1}{3}\pi r^2 h = \pi \approx 3.1416. A regular tetrahedron of edge 1 has volume \tfrac{1}{12}\sqrt{2} \approx 0.1179. If your measured number matches to three or four figures, the construction is placed as you intended.

The eccentricity of a section ties directly to the angles: e = \cos\beta / \cos\alpha when \beta is measured from the axis. For the worked ellipse that is e = c/a = 0.2966/1.264 = 0.2346, a fairly round ellipse. Push the plane shallower toward \beta = \alpha and e climbs to 1, where the ellipse opens into a parabola.

Common mistakes

A second trap is the ground-plane click. Clicking empty space drops a point on z = 0, not floating in mid-air. To put a point at height h, use point-by-coordinates with (0, 0, h), or drop a ground point and hold Shift to drag it vertically.

A third: the hyperbola looks incomplete. It is not. The board samples 240 generators and clips where they leave the drawn cone, so each branch simply stops at the edge of the model. Lengthen the cone and more of the branch appears.

Finally, do not read a relation-checker pass as a proof. When it says two planes stay perpendicular under random legal moves, it has tried many configurations and found no counterexample. That is a strong hint to go prove it, not the proof itself.

Related tools

If you want the flat version of this board, drag points and circles on the Dynamic Geometry Board. To spin the five perfect solids and check Euler's formula, try the Platonic Solids Explorer. For rotation in space by axis and angle, the Quaternion Rotation Visualizer reads off the quaternion, and the Tesseract Projection pushes the idea into four dimensions. To see how a matrix reshapes the plane before you lift into space, open the Linear Transformation Playground. And to plot functions of two variables as surfaces you can orbit, the 3D Surface Plotter uses the same camera.

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