Let's take a look at another application where UGFs are the appropriate setting: chamfers, in particular the constant-width flavor. (UGFs are a generalization of SDFs considering fields with unit gradient magnitude. www.blakecourter.com/2023/05/18/f...) Images are from #nTop . (1/n) - ThreadSky

bcourter.bsky.social • 94 days ago

Let's take a look at another application where UGFs are the appropriate setting: chamfers, in particular the constant-width flavor.

(UGFs are a generalization of SDFs considering fields with unit gradient magnitude. https://www.blakecourter.com/2023/05/18/field-notation.html)

Images are from #nTop .

(1/n)

Comments

bcourter.bsky.social•94 days ago

With this approach, you can create any surface in the local 2D coordinate system of the edge.

In the next session, we'll look at that two-surface coordinate system more closely, treating it as a basis for remapping any kind of edge treatment!

(10/n, n = 10)

bcourter.bsky.social•94 days ago

Instead, we can create a surface at an arbitrary angle Θ to the normalized S or D fields via S * cos(Θ) + D * sin(Θ) .

Algebraic geometers call this family a "pencil".

With such an angled face at any angle and offset, we can describe any surface from the edge in "Hesse normal form".

(9/n)

bcourter.bsky.social•94 days ago

But what about asymmetric chamfers? One approach is to expand A + B into

A * t + B * (1 - t)

Indeed, such interpolating will create suitable geometry, but how do we control it with CAD-like parameters?

(8/n)

bcourter.bsky.social•94 days ago

To convert to a constant width or a constant setback chamfer, one simply does some trig on the triangle to figure out what the inset should be based on the angle between the gradients of A and B.

(7/n)

bcourter.bsky.social•94 days ago

To get a chamfer, we just need to offset our normalized A + B field inward and intersect again. This result produces a constant inset chamfer, where the width increases with dihedral angle between faces.

(6/n)

bcourter.bsky.social•94 days ago

Here is A + B with A ∩ B overlaid above. Looks okay at first glance, but notice (left) that the spacing between the contours is wider than our original fields. It's gradient magnitude is not unity, but we can normalize the field by dividing by that magnitude (right).

(It's a bit subtle.)

(5/n)

bcourter.bsky.social•94 days ago

UGFs are more special than general implicits because their gradient has unit magnitude. The sum and difference of two unit vectors are perpendicular, as seen in the diagonals of the rhombus of the vectors.

Defining:
S = A + B
D = A - B

Interactive version: https://www.shadertoy.com/view/dd2cWy

(4/n)

bcourter.bsky.social•94 days ago

We will can intersect these together by taking the max of the fields. (Our sign convention is that inside is negative.)

This corner, A ∩ B, is what we want to chamfer. Note that this result is not an SDF, but a UGF, because the field extending up from the corner is sharp.

(3/n)

bcourter.bsky.social•94 days ago

Let's work in 2D so we can visualize the field. We will take two planes represented as signed distance fields (SDFs), and let's call those fields "A" and "B".

Here's A. B will just be it's mirror image across the vertical axis:

(2/n)

Comments

Posting Rules

Reply