Module elementary 

Parameterization of the analytic primitives.

Evaluation, derivatives and parameter inversion for every shape in crate::conic and crate::quadric. Each shape’s parameterization is defined relative to its own Frame, which is what makes a parameter value mean the same place across a save, a reload and a transform.

Conventions

• Circle, ellipse: angle from the frame’s x axis towards y.
• Hyperbola: (a cosh t, b sinh t), describing the +x branch.
• Parabola: (t^2 / (4 f), t) with the apex at t = 0.
• Plane: (u, v) are the coordinates along x and y.
• Cylinder: (angle, height).
• Cone: (angle, height), with the radius varying along the height.
• Sphere: (longitude, latitude), latitude in [-pi/2, pi/2].
• Torus: (angle about the axis, angle around the tube).

Every surface here follows the same (u, v) order, u going around and v going along. Getting that consistent matters: a caller that has to remember which surface reverses the convention will eventually forget.

Structs

CurvePoint

A point on a curve with its first two derivatives.

SurfacePoint

A point on a surface with its first derivatives.

Functions

circle_at

Evaluate a circle at angle.

circle_parameter

The angle of the point on a circle nearest p, in [0, 2*pi).

cone_at

Evaluate a cone at (angle, height).

cone_parameters

The (angle, height) of the point on a cone nearest p.

cylinder_at

Evaluate a cylinder at (angle, height).

cylinder_parameters

The (angle, height) of the point on a cylinder nearest p.

ellipse_at

Evaluate an ellipse at angle.

ellipse_parameter

The eccentric angle of the point on an ellipse in the direction of p.

hyperbola_at

Evaluate a hyperbola at t, on the +x branch.

hyperbola_parameter

The parameter of the point on a hyperbola in the direction of p.

line_at

Evaluate a line at t, measured in length along its direction.

line_parameter

The parameter of the projection of p onto a line.

parabola_at

Evaluate a parabola at t.

parabola_parameter

The parameter of the point on a parabola level with p.

plane_at

Evaluate a plane at (u, v), the coordinates along its x and y axes.

plane_parameters

The (u, v) of the projection of p onto a plane.

sphere_at

Evaluate a sphere at (longitude, latitude).

sphere_parameters

The (longitude, latitude) of the point on a sphere nearest p.

torus_at

Evaluate a torus at (around the axis, around the tube).

torus_parameters

The (u, v) of the point on a torus nearest p.

wrap_angle

Wrap angle into [0, 2*pi).

wrap_signed_angle

Wrap angle into (-pi, pi].