83 3D: Linear maps and rotations

9.3

Racket

83 3D: Linear maps and rotations🔗ℹ

struct
(struct linear3 (m00 m01 m02 m10 m11 m12 m20 m21 m22)
    #:transparent)
  m00 : finite-real?
  m01 : finite-real?
  m02 : finite-real?
  m10 : finite-real?
  m11 : finite-real?
  m12 : finite-real?
  m20 : finite-real?
  m21 : finite-real?
  m22 : finite-real?

A 3×3 matrix in row-major order, acting on column vectors. Thus (linear3 a b c d e f g h i) represents rows

[a b c]
[d e f]
[g h i]

and (linear3-compose outer inner) means outer ∘ inner: inner acts first.

value
identity-linear3 : linear3?

The identity matrix.

procedure
(linear3-compose outer inner) → linear3?
outer : linear3?
inner : linear3?

Composes two maps as outer ∘ inner.

procedure
(linear3-invert map) → linear3?
map : linear3?

Returns the inverse or raises an exception for a singular matrix.

procedure
(linear3-determinant map) → finite-real?
map : linear3?

Returns the determinant.

procedure
(linear3-transpose map) → linear3?
map : linear3?

Returns the transpose.

procedure
(linear3-apply-vector map value) → vec3?
map : linear3?
value : vec3?

Applies map to a column vector.

procedure
(linear3-normal-transform map) → linear3?
map : linear3?

Returns the inverse-transpose normal map. The linear map must be invertible.

procedure
(rotation3? value) → boolean?
value : any/c

Recognizes a normalized proper rotation.

procedure
(rotation3-components rotation) → vector?
rotation : rotation3?

Returns a debugging vector containing scalar-first quaternion components. The constructor is intentionally not public; all public construction normalizes and canonicalizes quaternion sign.

value
identity-rotation3 : rotation3?

The identity rotation.

procedure
(axis-angle axis angle) → rotation3?
axis : vec3?
angle : finite-real?

Constructs the right-handed rotation by angle radians about nonzero axis.

procedure
(rotation3-from-to from-direction
to-direction) → rotation3?
from-direction : vec3?
to-direction : vec3?

Returns the shortest rotation sending one nonzero direction to the other. Opposite directions use a deterministic perpendicular axis.

procedure
(rotation3-look-at forward [#:up up]) → rotation3?
forward : vec3?
up : vec3? = y-axis3

Maps local positive z to forward while keeping local positive y as close as possible to up. Parallel inputs are rejected.

procedure
(rotation3-compose outer inner) → rotation3?
outer : rotation3?
inner : rotation3?

Composes rotations as outer ∘ inner.

procedure
(rotation3-invert rotation) → rotation3?
rotation : rotation3?

Returns the inverse rotation.

procedure
(rotation3-apply rotation value) → vec3?
rotation : rotation3?
value : vec3?

Rotates a displacement vector.

procedure
(rotation3->linear3 rotation) → linear3?
rotation : rotation3?

Converts a rotation to its proper orthogonal matrix.

procedure
(rotation3-slerp from to progress) → rotation3?
  from : rotation3?
  to : rotation3?
  progress : (and/c finite-real? (>=/c 0) (<=/c 1))

Interpolates along the shortest quaternion arc. Near coincident rotations use normalized linear interpolation. Progress zero and one return the exact input endpoint values.

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