How to calculate good-looking bezier control points in 3D?

Processing Beginners
1

I’d like to draw Bezier curves algorithmically, i.e. I’d like the control points to be functions of two data points. Assuming the following Bezier curve:

image

I imagine P0, P1, and P2 form an obtuse triangle where the angle at P1, say β, is somewhere around 115 degrees. How do I find point P1 given that points P0 and P2 are determined by my data?

2

Do you really mean 3D?

This assume your plane / floor is in x/z and height is y

P1.x = (P0.x + P2.x) / 2;  
P1.y =  P0.y - 140; 
P1.z = (P0.z + P2.z) / 2;
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3

Yeah, I mean 3D. I know my drawing is 2D but that’s just for simplicity.

Your points work in principle but depending on how far apart the points are, I sometimes get a very heavy curvature and sometimes almost no curvature at all. Is there a way to make point P1 to be a function of the distance of P0 and P2? This way, I hope, the angle/curvature will be maintained.

4

unfortunately the angle is not sufficient to determine the position of P1 even in 2D
https://www.mathsisfun.com/geometry/circle-theorems.html

so you need to set more constraints. for example, P0P1 == P1P2. But again this is not sufficient in 3D. To deal with these situations, for example, Catmull-Rom uses adjacent points as control points (implemented as curveVertex in processing)

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5

Ah perfect, thanks! This is closer to what I’m looking for. Taking my 2D example from above, how could one create a line that just curves ever-so-slightly from a straight line between P0 and P2?

I have tried the following:

beginShape();
curveVertex(P0.x, P0.y, P0.z);
curveVertex(P2.x, P2.y, P2.z);
endShape();

I get curved lines from point P0 to P2 but they are bent very strongly and I’m wondering how I can tune down the curvature a bit, again not using a constant but by imputing a helpful in-between point (e.g. P1) that reduces the curvature.

6

I haven’t tried but curveTightness might do the job:
https://processing.org/reference/curveTightness_.html

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7

I came across it in this exact moment. It does do the job! Apparently, the apparent tightness depends on how far the points are apart. So I’m experimenting with 1/distanceBetweenPoints, 1/distanceBetweenPoints^2, etc. It seems to be working well.