OBSERVATION CHAMBER / ACTIVE
Pascal Projective Geometry Observatory
Move six points on a conic and apply a projective transformation. Observe Pascal's collinear intersections, including points at infinity, and compare an off-conic case.
101 / GEOMETRY / PROJECTIVE INCIDENCE
Six points. Three intersections. One line.
Cyan points lie on the transformed conic. Colored lines extend opposite sides; amber X, Y and Z lie on the Pascal line.
Drag a point along the conic, or select A–F and use the angle slider. With the canvas focused, ← / → changes the selected angle by 1°. The viewport stays fixed during dragging.
Canvas 2D is unavailable. Controls, coordinate tables, checks and reports still work. PNG is unavailable.
HOMOGENEOUS COORDINATES
Read finite and infinite points
- Construction validity
- Normalized collinearity residual (limit 10⁻¹⁰)
- Pascal line [a,b,c] : ax + by + cw = 0
| Point | θ | [x,y,w] | Location |
|---|
| Point | Definition | [x,y,w] | Location |
|---|
Homogeneous vectors have unit Euclidean norm. A finite point may be outside the viewport. INFINITY means |w| ≤ 10⁻¹² after normalization; it is not replaced with a large finite coordinate. The line at infinity itself has no finite segment to draw.
Incidence diagnostics
The exact BigInt fixture checks an integer rational parametrization. Floating checks test this implementation; small residuals are numerical agreement, not a formal proof.
Save the observation
Observation report
Each edit clears old output and checks. A coincident or undefined construction disables checks, REPORT and PNG.
OBSERVATION GUIDE / GEOMETRY
What survives a projective transformation?
- Start with IRREGULAR. Find X, Y and Z on the amber line. Select a point and change its angle.
- Change p and s. The circle becomes another nondegenerate conic; incidence and collinearity survive. At |p| = 1 the affine view is parabolic, and at |p| > 1 it is hyperbolic.
- Try PARALLEL to see intersections at infinity. Try COINCIDENT to remove a required hypothesis. OFF-CONIC is outside the theorem: the residual is generally nonzero, but failure is not guaranteed for every arrangement.
P(θ) = [cos θ, sin θ, 1]; H = [[1,s,0],[0,1,0],[p,0,1]]; det H = 1.
X = (A × B) × (D × E); Y = (B × C) × (E × F); Z = (C × D) × (F × A).
r = |det(X,Y,Z)| / (‖X‖ ‖Y‖ ‖Z‖).
Pascal’s theorem concerns six distinct points on a nondegenerate conic. All joins and meets use homogeneous cross products. This model observes projective incidence, not cubic-curve group addition, hyperbolic distance or complex mapping. It makes no claim that distances or angles are preserved.
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OBSERVATION POLICY
This homogeneous-coordinate projective model runs in your browser. Points, calculations, reports, and images stay local. It preserves incidence, not Euclidean distance, and does not measure real objects.
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