A priori error estimates for Lagrange interpolation on triangles

We present the error analysis of Lagrange interpolation on triangles. A new a priori error estimate is derived in which the bound is expressed in terms of the diameter and circumradius of a triangle. No geometric conditions on triangles are imposed in order to get this type of error estimates. To derive the new error estimate, we make use of the two key observations. The first is that squeezing a right isosceles triangle perpendicularly does not reduce the approximation property of Lagrange interpolation. An arbitrary triangle is obtained from a squeezed right triangle by a linear transformation. The second key observation is that the ratio of the singular values of the linear transformation is bounded by the circumradius of the target triangle.