20.1 Automatic Closed-Loop Morph Correspondence🔗ℹ

Closed-loop alignment combines path reversal and cyclic-start selection with the normalized morph engine. Given two single closed loops, align the destination explicitly with:

(define aligned-destination

(path-geometry-align-for-morph source destination))

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

(path-geometry-normalize-for-morph source aligned-destination))

The alignment search uses total-arc-length point samples. It considers both traversal directions by default, scores cyclic phases deterministically, and prefers forward traversal on exact ties. Set #:allow-reverse? #f when orientation is semantically significant and only a cyclic phase may change.

For ordinary animation, the combined request is shorter:

(scene-play scene

(morph-to-aligned panel destination)

#:duration 3)

The exact requested destination representation is still installed at eased progress one. Automatic alignment is an interior correspondence choice; it does not rewrite the caller’s endpoint object permanently.

This operation handles one positive finite closed subpath per side. Equal-count compound pairing is available separately, while topology-changing morphs add controlled subpath birth/death when topology-class counts differ.

Render the canonical comparison example with:

20.2 Automatic Open-Path Morph Correspondence🔗ℹ

Open-path alignment handles the endpoint-direction ambiguity of one open path. Prepare an explicit destination traversal with:

(define aligned-destination

(path-geometry-align-open-for-morph source destination))

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

(path-geometry-normalize-for-morph source aligned-destination))

The two paths must each contain exactly one positive finite open subpath. The algorithm samples corresponding total-arc-length fractions from both endpoints and compares the stored destination traversal with its semantic reversal. A strictly lower reverse score selects reversed correspondence; an exact tie keeps the stored destination direction. There is no cyclic phase search because open endpoints remain distinct.

For timeline use:

(scene-play scene

(morph-to-open-aligned panel destination)

#:duration 3)

The aligned/reversed representation is interior correspondence only. Eased progress zero uses the exact clip-start source path, and eased progress one installs the exact destination object originally requested by the caller.

Render the canonical comparison example with:

20.3 Automatic Compound-Path Morph Correspondence🔗ℹ

Compound closed-loop alignment handles paths whose subpath storage order differs. Prepare explicit geometry with:

(define aligned-destination

(path-geometry-align-compound-for-morph source destination))

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

(path-geometry-normalize-for-morph source aligned-destination))

Every source/destination loop pair is scored with closed-loop phase/direction correspondence. The procedure then chooses one global minimum-total-cost assignment, so a locally attractive early match cannot force a worse overall pairing. Pairing is deterministic and independent of rendering, camera state, frame rate, wall-clock time, or randomness.

For timeline use:

(scene-play scene

(morph-to-compound-aligned panel destination)

#:duration 3)

The source and destination must contain equal nonzero counts of positive finite closed subpaths. The exact requested destination representation, including its original subpath order, is installed at eased progress one. Reordering and per-loop alignment are interior correspondence only.

Render the canonical comparison example with:

20.4 Automatic Open-Compound Morph Correspondence🔗ℹ

Open-compound alignment applies endpoint-direction correspondence to equal-count compound figures whose subpaths are all open. Prepare explicit geometry with:

(define aligned-destination

(path-geometry-align-open-compound-for-morph source destination))

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

(path-geometry-normalize-for-morph source aligned-destination))

Every source/destination open-subpath pair is scored with inclusive total-arc-length forward/reverse correspondence. A global minimum-total-cost assignment then uses the same deterministic assignment policy as closed compound alignment. A locally attractive early match therefore cannot force a worse total correspondence.

For timeline use:

(scene-play scene

(morph-to-open-compound-aligned panel destination)

#:duration 3)

Pairing and per-pair reversal are interior correspondence only. Eased progress zero uses the exact clip-start source path, and eased progress one installs the exact caller-requested destination, including its original subpath order and stored directions.

Render the canonical comparison example with:

The upper panel uses stored-order normalized correspondence and cross-pairs the open curves. The lower panel pairs subpaths globally and chooses each endpoint direction independently.

20.5 Automatic Mixed-Topology Compound Morph Correspondence🔗ℹ

Mixed-compound alignment combines the open and closed correspondence rules for one compound path that may contain both topology classes. Prepare explicit geometry with:

(define aligned-destination

(path-geometry-align-mixed-compound-for-morph source destination))

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

(path-geometry-normalize-for-morph source aligned-destination))

The source and destination must be nonempty and every subpath must have positive finite arc length. Open counts must match independently from closed counts. Open candidate pairs use forward/reverse endpoint correspondence and closed candidate pairs use cyclic phase/direction correspondence. The same deterministic global assignment policy is solved separately inside both classes, then the selected destination subpaths are restored to source order.

For timeline use:

(scene-play scene

(morph-to-mixed-compound-aligned panel destination)

#:duration 3)

Topology pairing/reordering and per-pair alignment are interior correspondence only. Eased progress zero uses the exact clip-start source, and eased progress one installs the exact caller-requested destination, including its original interleaving and storage order.

Render the canonical comparison example with:

Every intended destination counterpart also changes geometry. The upper panel uses normalized stored-order correspondence and sweeps identities across the scene. The lower panel solves open and closed identities independently, so the green paths visibly morph while staying with their spatially sensible counterparts.