POV-Ray : Newsgroups : povray.binaries.images : Hesse normal form for lines - distance formula. Server Time
14 May 2021 16:11:20 EDT (-0400)
  Hesse normal form for lines - distance formula. (Message 1 to 1 of 1)  
From: William F Pokorny
Subject: Hesse normal form for lines - distance formula.
Date: 4 Dec 2020 10:36:52
Message: <5fca5794$1@news.povray.org>
The Hesse or Hessian normal form is often used to define planes. There 
is a similar encoding for lines; in an old book I stumbled across a 
distance formula used as a part the line encoding which makes for a nice 
method to specify planes by rotation and offset relative to the y axis.

It naturally divides regions into positive and negative values with good 
gradients.

// Set up for left handed rotation about y. 0 angle aligns with x.
#declare FnHessianDist = function (x,y,z,Theta,Offset) {
     z*cos(radians(Theta)) + x*sin(radians(Theta)) - Offset
}

It's useful. I'll be creating an inbuilt version called f_hessian_dist 
for povr. The SDL written function above can be used in any version of 
POV-Ray supporting functions.

Attached in an isosurface doodle making multiple uses of it all bound by 
a sphere. You can do octagonal columns and such too. Suppose other ways 
to do the same things, but being able to specify just angle and offset 
makes it easier.

Bill P.


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