2D Animation Principles

The keyframes are dialled in and the character still reads as stiff, because secondary motion is missing — the parts dragged along by the primary action that arrive a beat late. This covers parameterising a damped spring as frequency and damping ratio, and five traps in wiring it into a 2D skeletal system: why explicit Euler is unconditionally unstable, frame-rate dependent results, root-to-tip chain ordering, springing angles rather than positions, and why teleports blow it up.

Wire squash and stretch into a skeletal system and nearly all the difficulty comes from one fact: non-uniform scale does not propagate down a hierarchy as safely as rotation and translation. This covers area preservation in 2D and the two things it does not preserve, why the wrong pivot sinks a ball into the floor, and the core trap — a parent scale times a child rotation contains shear, so only the rotated parts come out skewed.

Publishing compiles the timeline into a CreateJS display tree, and the canvas repaints entirely every frame. Why vectors pay rasterisation per frame, the three cache parameter traps, and why filters require caching.

Linear blend skinning cannot express folds, draw order or follow-through. Why mesh deformation must be applied in bind space, why draw order is a discrete track, and how Live2D’s parameter-driven keyforms produce pseudo-3D.

Derive the frame-resolution formulas behind Graphic symbol loop modes, explain why a Movie Clip shows only frame 1 while authoring and collapses during frame-by-frame export, and why lip sync requires Graphic.

Shape tween explosions come from start-offset ambiguity on closed contours. Derive the correspondence cost function, show how shape hints act as constraints, and map it onto modern SVG path morphing.

Use derivatives to compare linear and eased velocity, then derive smoothstep, cubic Bezier curves, and frame-rate-independent timeline sampling.

Solve 2D shoulder and elbow angles analytically with the law of cosines while handling unreachable targets, numeric boundaries, and elbow continuity.

Derive linear blend skinning and explain normalized weights, inverse bind matrices, joint collapse, alpha contamination, and SVG seam dots.

Derive 2D forward kinematics from homogeneous matrices, pivot rotation, and parent-child transforms, then verify end-point coordinates interactively.