Rational Bézier Curves

Discusses how to think about and work with rational Bézier curves as projections of polynomial Bézier curves.

[kassemanis]

I have an exam in CAGD tommorow, i am trying to understand nurbs and the thing is, this video is so far the best on youtube. I hope to see more content like this

[policedog]

Thank you! Yours is the clearest of the dozen or so of Bezier or De Casteljau videos I've seen so far. One approach for demonstrating Bezier curves that I haven't seen anyone do a video of yet would be the following: Assemble three Bezier curves together to form a (two-dimensional) tri-lobed (not rectangular) shape that seamlessly closes on it's own origin. And each "lobe" could have its own curvature; which might be the same or might be different than the other two "lobes" at the whim of the designer. Of course with the constraint that the end result of the three curves must close on the origin. In other words if I could use your formulas to steer that 2D shape all the way around and have it smoothly meet up with itself without leaving a visible corner I think I would feel it to be a more complete or easier-to-remember mental model of the formulas. I'm still looking so if I find such a demo I'll let you know. So far all of the demos just have the end point of the curve trailing off into space.

[amoghrijal]

thank you! You are such a good teacher!! I think you should create more videos

[abhinavaggarwal]

Sir if it wasnt for ur video today i would have been forever lost in understanding probably the toughest part of my degree.. Cheers and please make more videos :D

[ShizhouLuo]

just confused about that why do we have to create the homo-coordinates?
is it because that we can improve the efficiency of our code when representing the bezier cure in a non-rational way(or say polynomial)?
hope someone can figure it out.
Truly thanks.

[holygl]

Absolutely amazing explanation, thank you so much!

I have a question that I was hoping you could help me with:

For a Bézier curve with 2 end points; let's call them p1 and p2—each with a control point, let's call them c1 and c2 respectively and using with weightings p1w, c1w, c2w, p2w

Is it possible to calculate a new Bézier curve that satisfies this curve?

So for the following example, how would you calculate the new end points and control points:

p1 = 1, 1 and p1w = 0.5
c1 = 5, 2 and c1w = 1.2
c2 = 9, 0 and c2w = 1.0
p2 = 6, 7 and p2w = 2.0

...hope that makes sense :)

[rayfletcher]

So a polynomial bezier is 2 dimensional, and a rational bezier curve is a projection of that 2 d curve, correct?

[xdxxdy]

cool I will try to make some curves in rhino and reconstruct the math with grasshopper, what program do you use to do it so intuitive?