Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

As the title says, I want to compute the root mean square distance between two n-dimensional simplices. Say I have two surfaces $S$ and $S'$, the mean error is $$ d_m(S,S') = \frac{1}{|S|} \int\int_{p \epsilon S} d(p,S')dS\,. $$ where $|S|$ denotes the volume of $S$ and $d(p,S')=\underset{p' \epsilon S'}{\text{min}}||p-p'||_2 $. The root mean square error is the $$ d_{rmse}(S,S')=\sqrt{ \frac{1}{|S|} \int\int_{p \epsilon S} d(p,S')^2dS\ },. $$

Say I have a triangle $T_{i,j}=(x_{i,j},x_{i+1,j},x_{i,j+1})$ and associated errors $e_{i,j}, e_{i+1,j}$, and $e_{i,j+1}$. According to this paper, If we linearly interpolate between the error values, the integral of $e^2$ is $$ \frac{|T_{i,j}|}{6}[e_{i,j}^2 + e_{i+1,j}^2 + e_{i,j+1}^2 + e_{i,j}e_{i+1,j}+ e_{i,j}e_{i,j+1} + e_{i+1,j}e_{i,j+1}] $$

So my questions are (1) How is the third formula above derived? (2) how would I extend this to $n$-dimensions.

Edit I thought an image might help


The bottom triangle is $T_{i,j}=(x_{i,j},x_{i+1,j},x_{i,j+1})$, and the top triangle is a vertical distance of $e_{i,j}, e_{i+1,j}$, and $e_{i,j+1}$ from each point. The mean error would be the volume between the triangles. I don't know what the root mean square error is graphically.

Edit2 Forgot to stay that the error is linearly interpolated. Thanks, joriki.

share|cite|improve this question
Why didn't you tell us that the error is linearly interpolated in the paper? Your question makes it sound as if that expression is supposed to be the integral over $d(p,S')^2$. – joriki Nov 29 '12 at 8:27

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.