Implicit

Prologue

1. 0:10This corridor never ends. Yet the program holds no list of cubes, and not a single vertex is stored.
2. 0:16There's only a short formula. Ask it about any point in space, and it answers with one thing: the distance to the nearest surface.
3. 0:24Four numbers are all the formula holds.
4. 0:26This film is about what you can make, with nothing but that one answer.

IHow far to the edge

1. 0:34On a flat plane, a shape has no surface, so we ask how far it is to the edge.
2. 0:38Take a point's distance to the centre, minus the radius: positive outside; on the circle, exactly 0; and inside, negative. Drop the sign, and it's the distance to the circle.
3. 0:49The circle is where this function is 0: no list of points, just an equation. That's what implicit means, and it's the film's title.
4. 0:57Bottom right is a slice: values along the line through the centre. Plotted, they make a V, with a kink at the centre.
5. 1:05Points of equal value form level sets. Only the drawing moves; the values stay put.
6. 1:10Take the bright ring: every point on it is 40 from the circle. Push each point of the circle out by 40, and you get this ring.
7. 1:18Ring after ring, like wavefronts moving out.
8. 1:21Distance has a rule. Head the way the value climbs fastest: it changes by just as much as you move; at a slant, by less.
9. 1:29In any direction, it never changes faster than you move: the V's slopes are ±1; off-centre, they're flatter.
10. 1:37Both formulas are 0 on the same circle; the one on the right changes too fast, and breaks the rule. Distance is special: drop the sign, and it's exactly how far the edge is.
11. 1:47Draw a circle round any point, with its distance to the edge as radius. It's empty: no edge inside, touching just the nearest point.
12. 1:56Inside the shape, drop the minus sign for the radius.
13. 1:59There can be more than one nearest point, as the next chapter shows. This empty circle returns in 3D.

IICorners and the skeleton

1. 2:09A rounded rectangle has a formula too, with two parameters.
2. 2:13b is half the core's width and height; thicken it all round by r, and the corners round off. The finished half-size is b plus r.
3. 2:22At the start, b is 0: that's the circle from the last chapter. From circle to rectangle, only the parameters change; at every moment, it's the exact distance.
4. 2:32Around the corners, the outer level sets become arcs. Inside, past the corner radius, they sharpen to points on the bisectors.
5. 2:40On a corner's bisector, a point is equally close to both edges. Where the nearest point isn't unique, those points form the skeleton; inside, the fastest climb goes more than one way.
6. 2:51The slice's kinks all lie on the skeleton. A circle's skeleton shrinks to its centre: the kink in the last chapter's V.
7. 2:58Pull this front inward, like a fire lit all along the edge, burning in at a steady pace: the fire fronts meet along the skeleton, and die out together along the middle.
8. 3:08The skeleton is where fire fronts meet.
9. 3:10Circles and rectangles are convex: outside, each point has one nearest point, so their skeletons stay inside. Around the star's tip, the nearest point glides to the next edge.
10. 3:21At an inner corner, it's different: on the bisector, both edges are equally near; cross it, and the nearest point jumps. Now there's a skeleton outside too.
11. 3:31The star isn't convex, yet its value is still exact: it keeps the rule.

IIICombining shapes

1. 3:39The union of two shapes: at every point, take the smaller value; negative counts as smaller.
2. 3:46Outside both, that's the nearer shape; if either is negative, the point is inside.
3. 3:51Where they overlap, take the deeper one.
4. 3:53Outside the union, the minimum is exact. Inside, not always: this buried arc still counts, though it's no longer an edge, so the reading is too short.
5. 4:03Taking the smaller keeps the rule. Walk to the real edge, and the value must reach 0, no faster than you move: so it can only under-report.
6. 4:11Now the answer may be just a lower bound: at least this far.
7. 4:15Take the larger: the intersection. It's inside only where both are negative.
8. 4:20Inside, it's exact; outside, not always: past the tips, the level sets stay sharp, where the exact distance would round off. There, it can only under-report; and when we draw in 3D, rays travel outside.
9. 4:34Flip B's sign, and inside and outside swap. Intersect it with A, and B is cut out of A: the difference.
10. 4:42It's an intersection too, so outside, it can only under-report. Later, it's how a cube becomes a sponge.
11. 4:49Back to the union, and add a third circle.
12. 4:51Now round off the sharp seam: where the values differ by less than k, push the smaller one down a little. The shapes flow together like water drops; the edge itself bulges out.
13. 5:02Two circles are combined first, then the third; with rounding, the order matters a little.
14. 5:07Near the seam, it's no longer exact, yet it never changes faster than you move, so, measured to its own new edge, it can only under-report. Shrink k to 0: an ordinary union again.

IVOnto the screen

1. 5:22Back to three dimensions: keep the same formula, spin the circle about a diameter, and it's a sphere.
2. 5:28The scene is a union: floor, sphere and torus. Slice it through the camera: each point here still measures, in 3D, how far the nearest surface is.
3. 5:38Each pixel gets a ray: start at the camera o, head in direction u for a distance t. We want its first hit, but no faces are stored, only this function.
4. 5:48The empty circle is now an empty sphere: its radius is the value here, and no surface lies inside, so the ray can safely go that far, and never cross a surface.
5. 5:58Far from everything, steps are long; skimming a surface, they're short. Step by step it closes in, until it's close enough to call a hit. This is sphere tracing.
6. 6:09Why is it safe? Along this ray, values never change faster than you move: the curve stays in this wedge, and never reaches 0 before the step lands.
7. 6:19Take chapter III's difference: outside, it can only under-report. The ray only travels outside, so it's still safe: the spheres are smaller, and it takes more steps, but it never crosses the surface.
8. 6:31Intersection, difference and smooth union rely on this, and later, so will twisted space and the sponge.
9. 6:38This fan of rays is one column of pixels, down the middle of the screen. Most hit within a few steps; rays skimming a surface take dozens; rays that hit nothing in range give up.
10. 6:49A small enough value counts as a hit; going too far, or running out of steps, means giving up. Both are approximate stopping rules. The number of steps is information too.
11. 7:00Now every pixel gets its own ray, shaded by how many steps it took: quick hits are dark; brightest are the rays grazing outlines, or skimming the distant floor.
12. 7:10The outlines appear on their own: this picture shows the effort of tracing.
13. 7:15Ask which way the function climbs fastest: straight out of the surface. That's the surface normal, the way it faces; a sharp edge doesn't face just one way. Knowing that, we can work out how brightly it's lit.
14. 7:27Trace again, towards the light: if something blocks it, that's shadow. If it slips past, divide the gap by the distance travelled; the smallest ratio sets the shade: smaller is darker. It's an approximate soft shadow.
15. 7:42Step out a little, the way the surface faces. If the answer is much less than the step, something is close by: a crevice, so it's a little darker. That's approximate too.
16. 7:52Light, shadow, the darkness in crevices: all come from asking the same function.

VInfinitely many cubes

1. 8:00The prologue's formula has just one cube: the union of a frame and a bead, the smaller of two distances.
2. 8:07Cut space into cells, each L across. Fold every point back into one cell, and ask about the single object there.
3. 8:14No copy is ever made, or checked one by one: a query costs the same as for one cube, though rays may take a few more steps.
4. 8:22One cell gives the exact distance: the object is centred, stays inside its cell, and is symmetric with the grid, so no copy elsewhere is ever closer. That's the prologue's corridor.
5. 8:33Or twist space round the corridor's axis: the further along, the more it turns, winding the cells into a spiral tunnel.
6. 8:40Twisting stretches some directions, so the value can over-report: divide by the local stretch. But it grows with distance from the axis, so dividing alone isn't quite enough; shorten it a little more, as a margin, and it under-reports again.
7. 8:55That margin was checked numerically, for this scene only.
8. 8:59Each cell's number sets its own angle: a turning wave sweeps across the grid, yet the formula still has one object.
9. 9:06But a neighbour may now be nearer: asking one cell alone over-reports.
10. 9:10So one more limit: the gap to the cell face, plus the room a neighbour always leaves, caps each step.
11. 9:16That's another lower bound: steps are shorter, but still safe.

VIAt every scale

1. 9:23First, a cube: chapter II's formula, one dimension up. Take the distance to the central cross, flip its sign, intersect with the cube: a difference, leaving 20 of 27 blocks.
2. 9:35Then, as in chapter V, fold space into cells a third as wide: scale the coordinates up by three, divide the distance by three, and the same formula works again, carving a smaller cross in every block.
3. 9:48Keep going, layer by layer, and it's the Menger sponge. Like any difference, it can only under-report.
4. 9:54How many dimensions does it have? First, count the blocks: 20 remain.
5. 9:59Scale by three, and count the copies: a segment takes 3, a square 9, a cube 27. Each is 3 to the power of the dimension.
6. 10:08Carved without end, the sponge takes 20, between 9 and 27: by this count, about 2.73 dimensions.
7. 10:16Its volume tends to 0, while its surface area has no limit; stop at any layer, and it's still a solid.
8. 10:23The farther from the camera, the wider the patch each pixel covers; layers finer than that are skipped, and unfold as we get closer.
9. 10:31Skipping holes just fattens the shape; its field still under-reports, so rays still never cross the surface. Of infinitely many layers, only the visible ones are computed: the screen shows an approximation.
10. 10:44Take the endless sponge: a corner block is the whole thing, a third as wide; so it's the corner of a bigger sponge, three times as wide.
11. 10:52Shrink to a third about the corner, and each layer becomes the next finer one; every value is divided by 3.
12. 10:59So move three times closer, and the frame returns to where it began; you can zoom forever, computing only the layers you can see.
13. 11:06We also stage the larger sponge, the fog and the colours to match, so no seam shows.

Coda

1. 11:13Shapes combined, drawn onto the screen, copied infinitely many times, carved at every scale: not a single vertex listed, not one copy made.
2. 11:22Just ask one function: how far is the nearest surface? Even “at least this far” is enough: step that far, and you never cross a surface.
3. 11:31So nothing has to be built in advance: each point is answered only when asked.
4. 11:36Hidden where a function is 0: the shape is implicit.