Planes and spheres in 3D space – distance from center to plane
MathematicsVectors & Coordinate GeometryAges 17–18
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Vectors & Coordinate Geometry
In a rotatable Oxyz coordinate system, the plane (P): Ax + By + Cz + D = 0 and a sphere with center I and radius R are drawn together with the normal vector n = (A; B; C) and the perpendicular segment IH. The simulation computes d(I, (P)) with the distance formula and compares it to R to show whether the plane cuts the sphere in a circle (drawing that intersection circle, radius r = √(R² − d²)), is tangent to it, or misses it entirely. The 2D | 3D switch opens a true 3D view (three.js): a transparent sphere, the plane (P), segment IH and the intersection circle (or point of tangency), which you can rotate and zoom.
Lesson: Equation of a plane, equation of a sphere, distance from a point to a plane
What it shows
In 3D coordinates, the plane Ax + By + Cz + D = 0 has normal vector n = (A; B; C), and the distance from a point I(x₀; y₀; z₀) to it is d = |Ax₀ + By₀ + Cz₀ + D|/√(A² + B² + C²). Comparing d with the radius R of a sphere centred at I gives their relative position: if d < R, the plane cuts the sphere in a circle centred at H, the foot of the perpendicular from I, with radius √(R² − d²); if d = R, it is tangent at H; if d > R, they do not meet.
How to use
Set Coefficient A of the plane, Coefficient B and Coefficient C to fix the normal direction, then change Coefficient D and watch the plane slide parallel to itself. Choose a Center I of the sphere and adjust Radius R. Ask students to predict the relative position from d(I, (P)) in the info panel before checking the drawing. The 3D button opens a rotatable view of the sphere, plane and segment IH; Initial view restores the camera.
Parameters you can change
- Coefficient A of the plane -4–4
- Coefficient B -4–4
- Coefficient C -4–4
- Coefficient D -8–8
- Center I of the sphere (can also be adjusted with the sliders) I(0; 0; 0), I(1; 1; 1), I(2; −1; 1), I(−1; 2; 2)
- Radius R 0.5–4
Questions to explore
- Keeping A, B and C fixed, which values of D make the plane tangent to the sphere?
- When is the radius of the intersection circle equal to the radius R of the sphere?
- What role does point H play for the intersection circle, and how is segment IH related to the plane?
