October 1, 2009
No CommentsAn Overview of Bézier Curves
Description
A detailed tutorial on Bézier curves. Step by step tutorial including several examples of when and how to use Bézier curves for reference.
Overview
A Bézier curve is any parametric curve. They are extremely important in animation and computer graphic. Bézier curves can be linear, quadratic, and cubic. When Bézier curves are linear, they are expressed by the equation ![\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1 \mbox{ , } t \in [0,1]](http://s.wordpress.com/latex.php?latex=%5Cmathbf%7BB%7D%28t%29%3D%5Cmathbf%7BP%7D_0%20%2B%20t%28%5Cmathbf%7BP%7D_1-%5Cmathbf%7BP%7D_0%29%3D%281-t%29%5Cmathbf%7BP%7D_0%20%2B%20t%5Cmathbf%7BP%7D_1%20%5Cmbox%7B%20%2C%20%7D%20t%20%5Cin%20%5B0%2C1%5D&bg=ffffff&fg=000000&s=0 "\mathbf{B}(t)=\mathbf{P}_0 + t(\mathbf{P}_1-\mathbf{P}_0)=(1-t)\mathbf{P}_0 + t\mathbf{P}_1 \mbox{ , } t \in [0,1]")
![\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2 \mbox{ , } t \in [0,1].](http://s.wordpress.com/latex.php?latex=%5Cmathbf%7BB%7D%28t%29%20%3D%20%281%20-%20t%29%5E%7B2%7D%5Cmathbf%7BP%7D_0%20%2B%202%281%20-%20t%29t%5Cmathbf%7BP%7D_1%20%2B%20t%5E%7B2%7D%5Cmathbf%7BP%7D_2%20%5Cmbox%7B%20%2C%20%7D%20t%20%5Cin%20%5B0%2C1%5D.&bg=ffffff&fg=000000&s=0 "\mathbf{B}(t) = (1 - t)^{2}\mathbf{P}_0 + 2(1 - t)t\mathbf{P}_1 + t^{2}\mathbf{P}_2 \mbox{ , } t \in [0,1].")
![\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } t \in [0,1].](http://s.wordpress.com/latex.php?latex=%5Cmathbf%7BB%7D%28t%29%3D%281-t%29%5E3%5Cmathbf%7BP%7D_0%2B3%281-t%29%5E2t%5Cmathbf%7BP%7D_1%2B3%281-t%29t%5E2%5Cmathbf%7BP%7D_2%2Bt%5E3%5Cmathbf%7BP%7D_3%20%5Cmbox%7B%20%2C%20%7D%20t%20%5Cin%20%5B0%2C1%5D.&bg=ffffff&fg=000000&s=0 "\mathbf{B}(t)=(1-t)^3\mathbf{P}_0+3(1-t)^2t\mathbf{P}_1+3(1-t)t^2\mathbf{P}_2+t^3\mathbf{P}_3 \mbox{ , } t \in [0,1].")
