Villarceau Circles

A horizontal cut through a doughnut-shaped surface suggests concentric circles. Villarceau circles come from a different cut: an oblique plane through the center that is tangent to a regular ring torus at two distinct points. The complete section is a pair of intersecting circles. Their appearance in a perspective drawing can disguise their true shape; the Section plane view removes that projection.

Predict, rotate, and check the section

Open the Villarceau Circles module. The colored dots are points on C+ and C−; the hollow white dots mark the two tangencies. The drawing fits the current torus to the panel, so compare numerical values when changing its size.

1. Press Reset experiment. Check Major radius R = 5, Tube radius r = 3, Plane azimuth = 0 and Trace angle = 90. Predict d = 4 and the two point coordinates from the worked table, then compare the visible readouts below the drawing.
2. Set View to Section plane. The axes have equal scale. Locate the centers at u = ±3 and the two white intersections at u = 0, v = ±4; both circles have radius 5.
3. Move Trace angle to 0 and then 180. Check the pairs (8,0,0)/(2,0,0) and (−2,0,0)/(−8,0,0). The circles stay fixed while their marked points travel.
4. Return View to 3D view, Trace angle to 90, and set Plane azimuth to 90. A quarter-turn around z sends C+ to (−4,3,3) and C− to (−4,−3,3), while radius and inclination remain unchanged.
5. Set Plane azimuth back to 0 and Tube radius r to 4. Predict d = 3 and tilt 53.130102°. Check the points (4,3,4) and (−4,3,4). The circle radius remains 5 while the tube becomes thicker.
6. Set Tube radius r to 0.25, then Major radius R to 2. The ring is still regular. The gap d = √63/4 rounds to 1.984313 and the inclination arcsin(1/8) rounds to 7.180756°. The module excludes the zero-radius tube.
7. Press Reset experiment, then reduce Major radius R to 2 directly. Tube radius r becomes 1.75 in the same update, keeping a 0.25 gap between the radii. Check d = √15/4 ≈ 0.968246 and tilt ≈ 61.044976°. This automatic clamp prevents a horn or spindle torus.
8. Press Reset experiment; set Camera azimuth to 75 and Camera elevation to 45. Drag the canvas or focus it and press arrow keys. The coordinates stay fixed. Home or Reset view removes only pan, retaining these camera angles. Reset experiment also restores camera 30/25 and all geometry defaults.