Conic sections

Equations
cone x² + y² = (z tan α)², plane z = z0 + x tan β  (α half-angle of the cone, β tilt of the plane from the horizontal)
the cut, by azimuth φ around the axis:  z(φ) = z0 / (1 − q cos φ),   q = tan α · tan β   - this q is the eccentricity of the curve's SHADOW on the floor
the curve's own eccentricity:  e = sin β / cos α   (the plane is foreshortened by cos β along the tilt);  e = q only at e = 1
e < 1 ellipse, e = 1 parabola (β = 90° − α: the plane parallel to a generator), e > 1 hyperbola (two branches, one on each nappe)
Dandelin spheres:  centre on the axis at c = z0 / (1 ± s), s = sin α / cos β, radius |c| sin α;  each touches the cone along a circle and the plane at ONE point - a focus
directrix:  where the plane meets the plane of a sphere's tangency circle, z = c cos² α
measured along the drawn curve:  |PF| / dist(P, directrix) = e at every point;  PF₁ + PF₂ = 2a on the ellipse;  |PF₁ − PF₂| = 2a on the hyperbola;  b² = a²(1 − e²) with the TRUE e (and not with q)
30°/30°:  q = 1/3, e = 1/√3 = 0.5774, a = 3/4, b = 0.6124, foci 0.8660 apart = 2ae, spheres at 0.634 and 2.366;  30°/60°: e = 1, second sphere at infinity;  30°/70°: e = 1.085, second sphere at −2.165, |PF₁ − PF₂| = 2.227