4 ms·
The implicit form is what's used when rasterizing, because we have the opposite problem: we don't care about the value of a specific bezier function at specific
by Jasper_ 7y ago
The implicit form is what's used when rasterizing, because we have the opposite problem: we don't care about the value of a specific bezier function at specific time t, we care about whether the pixel at location x,y is on inside the path created by a number of bezier segments (but it's even more complicated, because of winding order). That can't be solved with the parametric form alone.
- romwell 7y agoIndeed, but the terminology used in mathematics is very clear. So perhaps add this snippet to the article? Another thing possibly of value: where Bezier curves come from - automotive design. The machines that were tracing these curves out needed to solve the forward problem (where should the tool head be at time t?), and the designers needed an easy way to change the shape of the curve. The Bezier splines were (and still are) a great solution to that problem. For that reason, it's easy to convert Bezier splines into G-Code for either plotter - or, these days - 3D printer. Also, sampling the parametric form does give you a polygon (or a polygonal mesh), which is easier to work with. But that's another can of worms.
- delhanty 7y ago>That can't be solved with the parametric form alone. That's not strictly true - you could choose a numerical algorithm like Newton-Raphson. 3D B-rep modelers will typically solve x,y,z on curve that way because many of the curves won't have an implicit form of any reasonable low degree. In fact, some of them won't have an implicit form at all.