# Curvature Dependent Treshold Fresnel Effect

While reading a paper titled Line Drawing from 3D Models I came across a unique technique on page 6 that creates a Fresnel like effect that would be cool to recreate in blender. I have attempted this myself, however nodes never have been my strong suit and I failed to replicate it. here is the relevant equation n · v < ε √ Kᵣ and Below, I've detailed the parts I believe I've correctly equated.

n = Geometry (Normal)

· = Vector Math (Dot Droduct)

v = Geometry (incoming)

ε = Threshold Value

I've found myself quite lost with Kᵣ and what it represents, If anyone could provide guidance on how I can achieve the effects, I would welcome the assistance.

• I'm not going to read the complete paper, but since you said you have read it: is this graphic the only time the variables $n$, $v$, $\varepsilon$ and $\kappa_r$ are ever mentioned in all these pages? //Edit: I just saw that below the image, the text says $\sqrt{\kappa_r}$ is the square root of radial curvature. So, I don't know how you calculate it, but that's what it is. Aug 15, 2023 at 7:04

Lets re-visit the variables:

$$\textbf{n}$$ = Geometry (Normal) Yes, the normal of the surface point.

$$·$$ = Vector Math (Dot Droduct) Yes.

$$\textbf{v}$$ = Geometry (incoming) I believe you mean the right thing. It's the view vector from the camera to the surface point.

$$\epsilon$$ = Threshold Value Yes.

$$\kappa$$ (small kappa) the common character for curvature. There are different definitions of curvature, but in general it describes how strongly the surface is "not flat" at some surface point. Depending on the complexity of the definition one can express properties like convex/elliptic ($$\kappa > 0$$), concave/hyperbolic ($$\kappa < 0$$) and flat ($$\kappa = 0$$), see Gaussian curvature for example. The radial curvature is never negative ($$\kappa_r \geq 0$$) because the radius of a sphere (circle in 2D), which approximates the surface point's surrounding is used. For a flat spot the sphere would have infinite radius, thus $$\kappa_r = \frac{1}{r = \infty} = 0$$. For a very bend surrounding the sphere would be very small and $$\kappa_r \gg 0$$.

To implement this effect in Blender, i found this very nice post which sets the curvature value of every vertex as its weight. These values can then be accessed in shaders ultimatively, like demonstrated in another very nice post. Use the Camera Data and Geometry node to get $$\textbf{v}$$ and $$\textbf{n}$$ in the shader. With all the data at hand, you should be able to implement the formula and achieve the effect. But well, this will be a bit of work.