# Fourier integral over elements

Suppose I have a triangular element with vertices $${\vec{r_1},...,\vec{r_3}}$$ and a function $$f(\vec{r})$$. I want to calculate the fourier integral over this triangle such that:

$$F(k_x,k_y)=\int \int f(x)e^{-j2\pi(k_xx+k_yy)}dxdy \, .$$

Normally, for any integral inside such a triangle element, I would do the master element transformation and calculate the fourier transform as:

$$F(k_x,k_y)=\int_{0}^{1} \int_{0}^{1-\eta} f(x(\xi,\eta),y(\xi,\eta))e^{-j2\pi(k_xx(\xi,\eta)+k_yy(\xi,\eta))}|J|d\xi d\eta \, ,$$

where $$|J|$$ is the Jacobian of the transform from $$(x,y)$$ to $$(\xi,\eta)$$. In this case, I do not know if this approach is correct. Is there anyone that has used such an approach to calculate the Fourier transform of a "meshed" object before? I would also be very happy if someone could share an article on the subject.

• Does your triangle have 4 vertices? Oct 25 at 0:53
• There is nothing magic about the transformation to the reference element. It is simply doing a coordinate transformation under the integral. As a consequence, your formula looks correct to me and is a valid way to compute the integral. Oct 25 at 4:06