9.3
108 3D: Schlegel diagrams and polyhedral nets
3D reference map · First 3D picture
struct
(struct schlegel-diagram3d-data ( outer-face viewpoint plane vertex-positions edges face-polygons mappings diagnostics) #:transparent) outer-face : any/c viewpoint : vec3? plane : plane3? vertex-positions : vector? edges : vector? face-polygons : vector? mappings : hash? diagnostics : hash?
Prepared
Schlegel-projection data. The diagnostics retain the deterministic outer face,
canonical target-plane basis, margin, and viewpoint distance.
procedure
(prepare-schlegel-diagram3d complex [ #:outer-face selector #:viewpoint-distance distance #:margin margin #:tolerance tolerance]) → schlegel-diagram3d-data? complex : polyhedral-complex3d? selector : any/c = #f distance : positive-real? = 3 margin : nonnegative-real? = 1/20 tolerance : nonnegative-real? = 1e-8
Projects a selected simple outer face into a
canonical plane. It rejects projections whose inner vertices do not lie in the
outer polygon instead of returning a misleading diagram.
procedure
(schlegel-diagram3d data [ #:id id #:color color #:width width]) → spatial-visual? data : schlegel-diagram3d-data? id : symbol? = 'schlegel color : any/c = "slateblue" width : positive-real? = 2
Builds the flat 3D edge visual from prepared data.
struct
(struct net-overlap3d ( first-face second-face area boundary-crossings) #:transparent) first-face : exact-nonnegative-integer? second-face : exact-nonnegative-integer? area : nonnegative-real? boundary-crossings : exact-nonnegative-integer?
A positive-area pairwise overlap in root-plane
coordinates. Sharing only a hinge boundary has zero area and therefore does not
produce this record.
struct
(struct net-hinge3d (parent child primal-edge) #:transparent) parent : exact-nonnegative-integer? child : exact-nonnegative-integer? primal-edge : exact-nonnegative-integer?
One
oriented parent-to-child hinge in a face-adjacency spanning tree.
struct
(struct net-face-transform3d ( face source-origin source-e source-f source-n target-origin target-e target-f target-n) #:transparent) face : exact-nonnegative-integer? source-origin : vec3? source-e : vec3? source-f : vec3? source-n : vec3? target-origin : vec3? target-e : vec3? target-f : vec3? target-n : vec3?
An orientation-preserving rigid
frame from a source face to the net’s root plane.
struct
(struct polyhedron-net3d ( root-face hinge-tree cut-edges face-transforms flat-polygons overlaps diagnostics) #:transparent) root-face : exact-nonnegative-integer? hinge-tree : vector? cut-edges : vector? face-transforms : vector? flat-polygons : vector? overlaps : vector? diagnostics : hash?
Prepared
net data. diagnostics records strategy, search completion, selected
tree edges, overlap total, boundary crossings, and flat-net diameter.
procedure
(prepare-polyhedron-net3d complex [ #:root-face root #:hinges hinges #:strategy strategy #:search-limit limit]) → polyhedron-net3d? complex : polyhedral-complex3d? root : exact-nonnegative-integer? = 0 hinges : (or/c #f (listof exact-nonnegative-integer?)) = #f
strategy : (or/c 'breadth-first 'depth-first 'minimum-overlap) = 'breadth-first limit : exact-positive-integer? = 1000
Prepares a rigid flat net. Explicit hinges
must form a connected spanning tree. The automatic breadth/depth policies use
source edge order. 'minimum-overlap enumerates candidate trees in that
order and minimizes total positive overlap area, then boundary crossings and
flat-net diameter. When its bounded search is truncated,
(hash-ref (polyhedron-net3d-diagnostics result) 'search-complete?) is
false and no optimality claim is made.
procedure
net : polyhedron-net3d?
Returns one stable direct-child symbol per polygonal face, in face-index order.
An explicit polygonal face identity is retained, so face 'front is a
child at '(net front); generated faces use deterministic
'face-N identities.
procedure
(polyhedron-net3d-group complex net #:id id [ #:face-materials materials]) → group3d? complex : polyhedral-complex3d? net : polyhedron-net3d? id : symbol? materials : (or/c #f vector?) = #f
Creates the canonical group of independent mesh children
for a prepared net. materials, when present, has one material3d
per face and is useful for making a net’s face boundaries visually legible.
procedure
(polyhedron-net3d-sample-transforms net progress) → vector? net : polyhedron-net3d? progress : (real-in 0 1)
Returns the source-to-sampled local transform for every
canonical face child. Interior samples recursively rotate each child around its
source hinge and inherit its parent’s current rigid map; the root stays fixed.
The endpoints use exact identity and prepared-flat transforms.
procedure
(unfold-polyhedron3d target net) → any/c
target : spatial-path? net : polyhedron-net3d?
Returns a scene-play request that unfolds the direct
face children below target.
procedure
(fold-polyhedron3d target net) → any/c
target : spatial-path? net : polyhedron-net3d?
Returns the inverse request. It requires the exact flattened
endpoint produced by unfold-polyhedron3d, and returns the exact source
transforms.
The canonical runnable demonstration is "examples/3d/fold-unfold-polyhedron.rkt".
Limitations. Net faces must be simple polygonal faces. The overlap kernel handles simple concave polygons by deterministic ear triangulation, but does not repair self-intersection or hole boundaries. Fold/unfold presently requires the independent, identity-local-transform face meshes made by polyhedron-net3d-group; it does not yet retarget arbitrary authored mesh trees, animate labels/strokes with their faces, sequence hinges to avoid intermediate collisions, or solve a global collision-free folding path.