Understanding Quadratic Bézier Curves

http://antoineleclair.ca/2011/08/27/understandaing-quadratic-b%C3%A9zier-curves/

by antoineleclair • 15 years ago

72 16 15 years ago

antoineleclair.ca

CountHackulus• 15 years ago

Or written another way, it's a weighted average of weighted averages.

jacobolus• 15 years ago

Jason Davies's version using D3:

http://www.jasondavies.com/animated-bezier/

http://www.jasondavies.com/animated-bezier/animated-bezier.j...

waqf• 15 years ago

Thanks, I liked this better than the originally posted article, mostly because it explains higher order beziers.

tripzilch• 15 years ago

that was very clear and concise, thanks!

palish2• 15 years ago

Interesting... It looks like the speed of the dot isn't constant. I wonder if that would cause problems e.g. for animation purposes? (Like using the bezier curve as a way to animate the position of an object moving through 3D space in a game / in movie production...)

saucerful2• 15 years ago

Yes, if you want a constant speed you need to re-parametrize by arclength, which is straightforward for quadratic (or, more generally, polynomial) curves as there is a simple closed expression for their length.

palish2• 15 years ago

But those tend not to have a zero derivative at the vertices, no? Which would cause other animation problems, if you try to link two curves together.

curtis• 15 years ago

You are correct about the re-parametrization. A closed expression for arc-length is news to me however. Do you have a citation?

psykotic3• 15 years ago

Polynomial splines are pretty shitty compared to rational splines for animation and modeling, but they both have the problem of non-constant speed. It gets much worse with higher degree curves. You fix that by letting the animator separately specify a scalar speed curve so he can control where things speed up and down. To make this curve be natural to manipulate, it must be specified relative to an arc length parameterization. You can sometimes profitably combine the arc length reparameterization curve and the pre-integrated user-specified speed remapping into a single remapping curve that's baked out to disk, so the run-time curve evaluator only has to deal with a single level of parameter remapping on top of the raw spline.

This is just a fact of life with splines. They have non-uniform speed in their raw form and you have to deal with it. It's also why quaternion slerping being constant angular speed isn't a real advantage over lerp-and-normalize because they both trace out the same path and often you'll want to overlay a speed curve anyway (and lerp-and-normalize's speed is only barely non-uniform). Evaluate-and-normalize is the way to go for quaternions. It works great for multi-way skeletal animation blending and higher-order rotation interpolation (just do the spline evaluation in R^4 and project back onto the unit sphere). Casey has an old email that goes into some of this: http://mollyrocket.com/911.txt

cal5k2• 15 years ago

I think it's an interesting study in the jargon of specialization that I had to look up most of what you wrote in order to figure out that you weren't just making things up.

This happens in any specialization... An endocrinologist and a radiologist, for example, probably only speak half of the same medical language.

baddox• 15 years ago

They're just gifs, but wikipedia has some interesting animations.

http://en.wikipedia.org/wiki/Bezier_curve#Constructing_B.C3....

ontouchstart• 15 years ago

I did a few exercises with SVG in pure CoffeeScript (no external library)

Draggable Cubic Bézier curve: http://ontouchstart.posterous.com/explore-svg-with-coffeescr...

Cubic Bézier timing function CSS: http://ontouchstart.posterous.com/explore-svg-with-coffeescr...

Understanding Quadratic Bézier Curves

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