Splines on triangulations are functions that on each triangle can be represented as a bivariate polynomial and that join smoothly across edges shared between neighboring triangles. What does it mean to join smoothly?
Consider the figure nearby. It shows two triangles, a yellow one and a green one, that share the edge e. Let y and g denote the polynomials on the two triangles, respectively. Let A be a point on e and B a point in the interior of the yellow triangle. A point P on the blue line connecting A and B and crossing e can be written as
P = tB + (1-t)A.
Thus, when t=0 then P is the intersection of the common edge e and the blue line. If t is (small and) positive then P lies in the yellow triangle, if it is small and negative it lies in the green triangle. Now consider the function
y(P) if t > 0 s(t) = g(P) if t <= 0
As t ranges from a negative value through zero to a positive value the point P moves along the blue line from the green triangle across e into the yellow triangle. The two polynomials join continuously if the function s is continuous at t=0 for all possible choices of A and B. More generally, they join smoothly of order r if the function s is r times differentiable, again for all choices of A and B. Following are some more technical remarks: