"Martin" asks:
Suppose I have a shape defined as (x,y,z) -> Bool how can I find a Point inside this shape? Obviously I could iterate through all possible x,y and z, but this appears very expensive.
Janis Voigtländer comments :
What if it is additionally known that the shape represented by (x,y,z) -> Bool has a closed, convex area (for example)? Most likely there are then techniques from algorithmic geometry that can find an inside point more efficiently than by iterating blindly through all coordinate triples.
Am Donnerstag, 29. Oktober 2015 schrieb Tom Ellis :
... There is no better way in general, so if you want to find points inside a shape you should use a different encoding of shapes.
You might discourage Martin from using his encoding, suggest using something different, nevertheless /*some people NEED implicit surfaces*/, useful for many purposes (e.g. for the ray tracing; polygonizing them may be horrible...) I don't understand what does it mean "find a point". ANY point? There is no "clever" solution for this yes/no relation. But people in image synthesis use more treatable representation: surf :: Point -> Real (e.g. a sphere = x^2+y^2+z^2-R^2, and not: x^2+y^2+z^2-R^2 < 0 . ) where, say, the interior is negative, exterior positive. Then with some initial, perhaps random steps, you may search with the aid of a moving simplex, or similar. And if the function is reasonably decent, gradient methods are good. This permits to find interesting points, such as barycenters or the surface itself. Jerzy Karczmarczuk