[Math] Sample of a subset of a plane

Question

I have the equation of a plane $ax+bx+cx+d$ and a point $(x_0, y_0, z_0)$ on that plane.

I defined the neighborhood of that point on that plane as the set of points satisfying $(x-x_0)^2 + (y-y_0)^2 + (z-z_0)^2 \le \epsilon^2.$

(Am I right?)

Now I need to sample that neighbor to get some points: how can I do it?

Thanks in advance.

Answer 1

The easiest thing to do would be to represent the plane as an affine space, and calculate an orthonormal basis for it. Then, you could simply add to $(x_0,y_0,z_0)$ any linear combination of those two vectors with coefficients $a^2+b^2\leq \varepsilon^2$.

Hint: look for vectors which are perpendicular to the normal to the plane.