Doug Kerr
Well-known member
Paths, and the boundaries of vector masks, in Photoshop (and many other image editing programs) are defined as composed of cubic Bézier curves - even straight line segments are a special case of that.
A cubic Bézier curve is an open two dimensional curve that is precisely defined by the locations of four points, often said to be two anchor points (which are in fact the endpoints of the curve) and a control point associated with each anchor point. Thus the locations of those four points become a precise mathematical specification of the curve.
More complex curves (including closed curves) are made of multiple Bézier curves joined together. In that case, at each "joint", the two anchor points (being the endpoints of the segments of the entire curve) coalesce into one, and are attended by the two control points.
Formally, a curve composed of two or more individual Bézier curves joined together is called a Bézier spline, although you may often hear it (incorrectly) called a "Bézier curve".
If the joint is a "corner" (if the adjacent segments are not line segments, we may speak of it as a cusp), the two control points usually actually lie right at the joint (although they don't have to!).
I have just released a new technical article, "Bézier Curves and Splines", available here:
http://dougkerr.net/Pumpkin/articles/Bezier_Curves.pdf
It discusses this matter thoroughly, progressing through several stages: conceptual, intuitive, metaphorical, geometric, and mathematical.
Best regards,
Doug
A cubic Bézier curve is an open two dimensional curve that is precisely defined by the locations of four points, often said to be two anchor points (which are in fact the endpoints of the curve) and a control point associated with each anchor point. Thus the locations of those four points become a precise mathematical specification of the curve.
(In formal mathematical writing, all four points are called "control points", which can lead to considerable confusion.)
More complex curves (including closed curves) are made of multiple Bézier curves joined together. In that case, at each "joint", the two anchor points (being the endpoints of the segments of the entire curve) coalesce into one, and are attended by the two control points.
Formally, a curve composed of two or more individual Bézier curves joined together is called a Bézier spline, although you may often hear it (incorrectly) called a "Bézier curve".
If the joint is a "corner" (if the adjacent segments are not line segments, we may speak of it as a cusp), the two control points usually actually lie right at the joint (although they don't have to!).
I have just released a new technical article, "Bézier Curves and Splines", available here:
http://dougkerr.net/Pumpkin/articles/Bezier_Curves.pdf
It discusses this matter thoroughly, progressing through several stages: conceptual, intuitive, metaphorical, geometric, and mathematical.
Best regards,
Doug