99 3D: Parametric surfaces and calculus

9.3

Racket

99 3D: Parametric surfaces and calculus🔗ℹ

Parametric surfaces use a fixed rectangular sampling topology. A surface is sampled once at inclusive parameter-grid sites and therefore has stable vertex and triangle identities. Its normals use declared analytic derivatives when both are supplied; otherwise they use deterministic centred/one-sided finite differences, adjacent-face fallback, and explicit unresolved-index reporting.

procedure
(parametric-surface3d procedure
[ #:u-range u-range
#:v-range v-range
#:resolution resolution]
#:id id) → surface3d?
  procedure : procedure?
  u-range : list? = (list -1 1)
  v-range : list? = (list -1 1)
  resolution : list? = (list 33 33)
  id : symbol?

Creates a fixed rectangular sampled parameterization.

procedure
(function-surface3d function
[ #:x-range x-range
#:y-range y-range
#:resolution resolution]
#:id id) → surface3d?
  function : procedure?
  x-range : list? = (list -1 1)
  y-range : list? = (list -1 1)
  resolution : list? = (list 33 33)
  id : symbol?

Creates the graph (vec3 x y (function x y)) and retains scalar-field data for calculus helpers.

procedure
(surface3d? value) → boolean?
value : any/c

Recognizes an immutable sampled surface.

procedure
(surface3d-resolution surface) → list?
surface : surface3d?

Returns its fixed (list u-count v-count) topology.

procedure
(surface3d-position-at surface u v) → vec3?
  surface : surface3d?
  u : finite-real?
  v : finite-real?

Evaluates a point in the authored parameter domain.

procedure
(surface3d-normal-at surface u v) → vec3?
  surface : surface3d?
  u : finite-real?
  v : finite-real?

Returns a safe unit normal, using the recorded deterministic fallback when necessary.

procedure
(surface-color surface color) → surface3d?
surface : surface3d?
color : color-spec?

Changes the uniform material colour without changing samples or topology.

procedure
(surface-color-by-height surface) → surface3d?
surface : surface3d?

Adds an opaque per-vertex z-height colour field.

procedure
(surface-color-by-scalar surface scalar) → surface3d?
surface : surface3d?
scalar : procedure?

Adds a deterministic opaque per-vertex colour ramp from the scalar evaluated at each existing sample.

procedure
(surface-checkerboard surface) → surface3d?
surface : surface3d?

Adds a deterministic parameter-space checkerboard colour field.

procedure
(surface-point surface u v #:id id) → mesh3d?
  surface : surface3d?
  u : finite-real?
  v : finite-real?
  id : symbol?

Creates a point at a surface parameter.

procedure
(surface-tangent-u surface u v #:id id) → group3d?
  surface : surface3d?
  u : finite-real?
  v : finite-real?
  id : symbol?

Draws an arrow in the u tangent direction.

procedure
(surface-tangent-v surface u v #:id id) → group3d?
  surface : surface3d?
  u : finite-real?
  v : finite-real?
  id : symbol?

Draws an arrow in the v tangent direction.

procedure
(surface-normal surface u v #:id id) → group3d?
  surface : surface3d?
  u : finite-real?
  v : finite-real?
  id : symbol?

Draws an arrow in the direct tangent-plane normal direction.

procedure
(surface-tangent-plane surface u v #:id id) → mesh3d?
  surface : surface3d?
  u : finite-real?
  v : finite-real?
  id : symbol?

Creates a finite tangent parallelogram at the parameter point.

procedure
(surface-coordinate-curve surface #:id id) → curve3d?
surface : surface3d?
id : symbol?

Samples one fixed-u or fixed-v coordinate curve.

procedure
(surface-gradient-arrow surface x y #:id id) → group3d?
  surface : surface3d?
  x : finite-real?
  y : finite-real?
  id : symbol?

Creates the xy gradient arrow for a declared function-surface3d.

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