hr-studioGeometry & Structure · 04
A computational design primer · Subdivision

A coarse cage, refined to silk.

You don't have to draw a smooth surface — you can grow one. Start with a blocky cage of a few points, apply one small refinement rule, and apply it again, and again. Each pass splits and averages; the shape softens toward a limit that is perfectly smooth. This is subdivision — how film characters, car bodies and furniture are modelled from a handful of control points.

Scroll to begin
i.

Cut the corners

The simplest rule is Chaikin's: walk around the polygon and replace every corner with two points, one a quarter of the way along each adjoining edge. Every pass shaves the corners a little finer; after four or five the polygon has melted into a smooth curve — the limit is a quadratic B-spline that kisses the midpoint of every original edge.

Drag the cage points; slide the depth from raw to silk.

¼ and ¾ along every edge, repeated
ii.

Through the points, or near them

Chaikin's curve pulls inside its cage — it approximates. A different rule inserts each new point from four neighbours, weighted (−1, 9, 9, −1)/16, and keeps every original point untouched: the curve interpolates, threading exactly through your cage. The trade is character — approximating curves are rounder and calmer; interpolating ones follow orders and can overshoot.

Toggle the scheme and drag a point to feel the difference.

Approximating hugs inside · interpolating threads through
iii.

Keeping an edge sharp

Smooth everywhere is its own kind of boring. Tag a point as a crease and the rule leaves it alone while everything around it softens — a crisp corner surviving in a smooth body. Better still, hold the tag for only a few levels and then release it: the corner rounds with a tiny, controlled radius. That's the semi-sharp crease, straight out of Pixar's toolbox.

Tap points to cycle them: smooth → semi-sharp → sharp.

Tap a point: smooth · semi · sharp
iv.

The same idea, in 3D

Surfaces subdivide too. Catmull–Clark splits every face into four and averages three ways — face centres, edge midpoints, vertex neighbourhoods — and a hard-edged box swells into a smooth, even body. Two levels and the silhouette is already round; three and the facets all but vanish. Every Pixar character and most product surfaces begin life as a cage like this.

Slide the depth; the view turns slowly on its own.

Cube → sphere-soft body, one rule
v.

The cage sculptor

Now sculpt. The wireframe is your cage — grab any of its eight corners and pull, and the smooth body follows like a skin stretched over your intent. Coarse moves in, silk comes out: stretch it into a pillow, pinch it into a wedge, drag corners through each other and watch the surface negotiate. Orbit by dragging empty space, choose your depth, and show or hide the machinery.

Drag a corner ◦ to sculpt · drag space to orbit
Split, average, repeat — and a fistful of points becomes a surface with no facets left to see. Subdivision is the argument that smoothness isn't drawn; it's the limit of patient refinement.