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"Martin" asks:<br>
<blockquote type="cite"> Suppose I have a shape defined as<br>
(x,y,z) -> Bool<br>
how can I find a Point inside this shape? Obviously I could
iterate through all possible x,y and z, but this appears<br>
very expensive.</blockquote>
<br>
Janis Voigtländer comments :<br>
<blockquote
cite="mid:CAGkFuyA_wmjq76BH8tLqAYszEin+t_HxDx71KrUD09e0xCKpoA@mail.gmail.com"
type="cite">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.<br>
<br>
Am Donnerstag, 29. Oktober 2015 schrieb Tom Ellis :<br>
<blockquote class="gmail_quote" style="margin:0 0 0
.8ex;border-left:1px #ccc solid;padding-left:1ex">...<br>
There is no better way in general, so if you want to find points
inside a<br>
shape you should use a different encoding of shapes.<br>
</blockquote>
</blockquote>
You might discourage Martin from using his encoding, suggest using
something different, nevertheless <i><b>some people NEED implicit
surfaces</b></i>, useful for many purposes (e.g. for the ray
tracing; polygonizing them may be horrible...)<br>
<br>
I don't understand what does it mean "find a point". <br>
ANY point?<br>
<br>
There is no "clever" solution for this yes/no relation. But people
in image synthesis use more treatable representation:<br>
surf :: Point -> Real<br>
<br>
(e.g. a sphere = x^2+y^2+z^2-R^2, and not: x^2+y^2+z^2-R^2 < 0
. )<br>
<br>
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.<br>
<br>
Jerzy Karczmarczuk<br>
<br>
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