21.1 Topology-Changing Morphs and Subpath Birth/Death🔗ℹ

Topology-changing morph preparation supports compounds whose open or closed subpath counts differ. Prepare explicit interior geometry with:

(define-values (prepared-source prepared-destination)

(path-geometry-prepare-topology-changing-morph source destination))

(define-values (normalized-source normalized-destination)

(path-geometry-normalize-for-morph

prepared-source prepared-destination))

Real open and closed subpaths are matched independently with the correspondence open-path and closed-loop correspondence rules. Under the default forced-only policy, if one class has extra real subpaths, only the forced count difference is unmatched. A born subpath grows from a one-segment degenerate seed at its destination bounds center; a dying subpath collapses to an analogous seed at its source bounds center. Explicit local anchor points can override that placement without changing the default, and numeric penalties can permit additional voluntary unmatched slots.

For timeline use:

(scene-play scene

(morph-to-topology-changing panel destination)

#:duration 3)

Prepared birth/death slots exist only in interior normalized geometry. Eased progress zero is the exact source object and eased progress one is the exact caller destination, so endpoint subpath counts and ordering remain exact.

Render the canonical example with:

The surviving open and closed paths visibly change shape, one unmatched open curve collapses, and one unmatched closed loop grows. Destination storage also scrambles surviving direction/phase so automatic correspondence remains visible in the same movie.

21.2 Explicit Birth/Death Anchors🔗ℹ

The #:birth-anchor and #:death-anchor options make seed placement configurable while leaving correspondence policy unchanged. Bounds-center behavior remains the default. To use one local hub for both sides:

(define hub (vec2 0 0))

(scene-play scene

(morph-to-topology-changing

panel destination

#:birth-anchor hub

#:death-anchor hub)

#:duration 3)

An explicit anchor is a point in the path Visual’s local coordinate system. The Visual’s translation, rotation, and scale apply afterward. The anchor is used only for unmatched subpaths; matched real correspondence remains governed by the open/closed geometric scoring and the existing topology-class assignments.

The only symbolic value accepted by either keyword is 'bounds-center. Each side has one shared anchor point. Numeric penalties may create additional voluntary unmatched slots. Sparse per-subpath anchor overrides select individual seed positions, while sparse numeric cost overrides independently influence which subpaths become unmatched.

Render the canonical example with:

The movie marks the explicit hub with a red diamond. One lower curve collapses into that hub while a new lower loop grows from it; a surviving upper curve visibly morphs at the same time.

21.3 Per-Subpath Birth/Death Anchors🔗ℹ

The per-subpath anchor maps keep the shared anchors as fallbacks and add sparse overrides for individual endpoint subpaths:

(define left-hub (vec2 -5 -1))

(define right-hub (vec2 5 -1))

(scene-play scene

(morph-to-topology-changing

panel destination

#:birth-anchor origin

#:death-anchor origin

#:birth-anchor-map (hash 1 left-hub 2 right-hub)

#:death-anchor-map (hash 1 left-hub 2 right-hub))

#:duration 3)

A birth-map key names the exact original subpath index in the caller’s destination. A death-map key names the exact original subpath index in the clip-start source path. Topology partitioning, global assignment, source-order reconstruction, destination reversal, and closed-loop phase alignment do not change those key meanings.

Each map value is either 'bounds-center or a finite local vec2. If a key is absent, the corresponding shared birth-anchor or death-anchor is used. An explicit 'bounds-center entry therefore lets one subpath use its own center even when the shared fallback is a custom hub. Matched real subpaths ignore anchor maps.

The same lookup applies to voluntary unmatched slots selected by numeric penalties. Anchor maps never alter real-pair costs or assignment policy. Timeline requests copy the maps into immutable hashes when the request is constructed, so later mutation of the caller’s input hash has no effect. Direct geometry preparation validates key ranges immediately; request compilation checks them against the clip-start source and stored destination.

Render the canonical example with:

The two lower source paths collapse into separate marked hubs while two new closed loops grow from those corresponding hubs. A surviving upper curve morphs at the same time.

21.4 Penalized Topology-Changing Correspondence🔗ℹ

The birth and death penalty options make forced real correspondence optional when the caller supplies an explicit cost for both sides of a replacement. The default remains unchanged:

(path-geometry-prepare-topology-changing-morph

source destination

#:birth-penalty 'forced

#:death-penalty 'forced)

To let the global matcher reject a poor source/destination pair, provide finite nonnegative local path-unit costs:

(scene-play scene

(morph-to-topology-changing

panel destination

#:birth-penalty 2

#:death-penalty 2)

#:duration 3)

A real edge keeps the existing geometric correspondence score. A death costs death-penalty, and a birth costs birth-penalty. The assignment is solved globally within each open/closed topology class. Thus a good nearby pair can remain matched while a distant pair is represented by one local collapse and one local regrowth. This is not a greedy threshold applied independently to each candidate.

The assignment minimizes two objectives lexicographically. First it minimizes total geometric/penalty cost. If that primary cost ties exactly, it minimizes the total number of birth/death edges. A real correspondence whose score equals death plus birth cost is therefore retained. Remaining exact ties use the same deterministic index ordering as the existing assignment machinery.

Both penalty keywords must use one mode: either both are the exact symbol 'forced, or both are finite nonnegative real numbers. Anchor selection is independent. Every unmatched slot—whether forced by a count difference or selected voluntarily by numeric penalties—uses the configured birth/death anchor policy. Synthetic slots remain interior correspondence only; exact source and caller destination representations are preserved at the clip endpoints.

Render the canonical comparison with:

The upper path uses forced correspondence and sweeps between distant locations. The lower path uses numeric penalties and instead collapses locally on the left while its destination grows locally on the right.

21.5 Per-Subpath Birth/Death Penalties🔗ℹ

#:birth-penalty-map and #:death-penalty-map keep the shared numeric costs as fallbacks and add sparse original-index overrides:

(scene-play scene

(morph-to-topology-changing

panel destination

#:birth-penalty 2

#:death-penalty 2

#:birth-penalty-map (hash 1 20)

#:death-penalty-map (hash 0 20))

#:duration 3)

birth-penalty-map keys address the caller destination’s original subpath indexes. death-penalty-map keys address the clip-start source’s original subpath indexes. Topology partitioning, global assignment, destination reordering, open-path reversal, and closed-loop phase selection do not renumber these keys.

Every map value is a finite nonnegative real cost. A missing key inherits the corresponding shared birth-penalty or death-penalty. Sparse values change only source-to-dummy death costs and dummy-to-destination birth costs in the augmented assignment. Real-pair geometric scores and the lexicographic exact-tie preference for fewer topology changes are unchanged.

Nonempty maps require numeric shared penalty mode; they are rejected when both shared penalties are 'forced. Anchor policy remains independent: The shared and per-subpath anchor settings determine where any unmatched subpath selected by the cost policy collapses or grows. Timeline requests copy penalty maps into immutable hashes so later mutation of caller hashes has no effect. Direct preparation validates map ranges immediately; request compilation validates them against the clip-start source and stored destination.

Render the canonical comparison with:

The upper panel uses shared low costs and replaces both distant pairs. The lower panel raises the death cost of original source index 0 and the birth cost of original destination index 1; that upper pair remains a real morph while the lower pair still collapses and regrows.

21.6 Per-Pair Real-Match Penalties🔗ℹ

#:match-penalty-map adds sparse, deterministic semantic costs to individual real assignment edges while retaining the geometric correspondence score:

(scene-play scene

(morph-to-topology-changing

panel destination

#:match-penalty-map

(hash (cons 0 0) 20

(cons 1 1) 5))

#:duration 3)

Each key is (cons source-index destination-index) in the original caller subpath order. The value is a finite nonnegative additive cost. Missing pairs add zero. Topology partitioning, global assignment, destination reordering, open-path reversal, and closed-loop phase selection never renumber the indexes.

The map affects real source/destination edges only. In forced mode it can change which same-topology real subpaths pair without enabling optional birth/death. In numeric penalty mode the increased real-edge score also competes with the resolved death and birth costs. Exact primary-cost ties retain the secondary preference for fewer topology changes.

Direct geometry preparation rejects out-of-range and open-to-closed keys. Timeline requests validate key shape/value at construction, snapshot the hash immutably, and validate range/topology at clip compilation. Anchor maps and endpoint penalty maps remain independent.

Render the canonical comparison with:

The upper panel uses geometry alone and both curves visibly morph locally. The lower panel penalizes original pair (cons 0 0), so the global assignment swaps the two real destination identities and the curves cross.