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Clarification on SAT?


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#1 Czar05   Members   -  Reputation: 1292

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Posted 08 June 2014 - 02:36 PM

Hi, I have been reading up on the Seperating Axis Theorem, and I have some questions.

 

http://www.codezealot.org/archives/55

 

The article does a great job on explaining what the SAT is, but I want to know what axes is the author referring to exactly. Would I use the X and Y axes or make my own arbitrary axes?

 

Also the author talks about "flipping the coordinates and negating one." in orderto get the normals for the edges. What did he mean by that?

 

Thanks in advance.



#2 Aardvajk   Crossbones+   -  Reputation: 11176

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Posted 08 June 2014 - 03:01 PM

The minimum set of axes you need to test are all those at right angles to the faces of the polyhedra. If none of these have an overlap the shapes are disjoint.

To get a left hand right angle vector for x, y, you simply use -y, x.

 

[EDIT - bit more info now not on phone :)]

 

Assuming 2D here, you can loop through each shape and construct a vector from each vertex to the next one (B - A, C - B etc where the vertices are in A, B, C... order). If you then do the -y, x trick you end up with vectors pointing outwards at right angles from each face. These vectors represent the axes you need to test by projecting the vertices of each shape onto them and then comparing the min-max of each projection to see if they overlap.

 

You can ignore any axes that are equivalent to those that are already being tested, including any that point in the exact opposite direction as well as those which point in the same direction. It is common to precalculate these axes so you can trim out these duplicates as a post-processing step.


Edited by Aardvajk, 08 June 2014 - 03:25 PM.


#3 Czar05   Members   -  Reputation: 1292

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Posted 08 June 2014 - 03:28 PM

Thanks for the response  Aardvajk,

 

Sorry I was not clear before, I meant to ask which axes would I project my shapes edge on the X  and Y axes or on an arbitrary axes that is made up by myself.


Edited by Czar05, 08 June 2014 - 03:29 PM.


#4 Burnt_Fyr   Members   -  Reputation: 1594

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Posted 08 June 2014 - 05:12 PM

Thanks for the response  Aardvajk,

 

Sorry I was not clear before, I meant to ask which axes would I project my shapes edge on the X  and Y axes or on an arbitrary axes that is made up by myself.

I think Aardvajk covered that pretty well. For 2 objects in 2D, you will need to project against each face normal of each object, with the exception of normals that are colinear with another normal. For a Triangle, you would have 3 axii, each perpendicular to one of the triangle's edges. For a square, you have 2 axii, the local X and Y, as the other 2 normals are -x and -y, which are collinear with their respective inverses.


Edited by Burnt_Fyr, 08 June 2014 - 05:12 PM.


#5 Aardvajk   Crossbones+   -  Reputation: 11176

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Posted 09 June 2014 - 01:35 AM

Let's see if a picture helps.

 

sat.jpg

 

The red lines are the normals of each of the faces of the shapes, so in this case you have the three for the triangle and the two red for the square. You should be able to understand that the two green lines are not also required as they are directly opposite the existing red lines for the square.

 

So the set of axes to test is the total set of face normals of the two shapes, less any that are the same as or directly opposite the ones already being tested. It can be proven mathematically by more clever bods than I that if these axes are all without overlap, there exist no axes with an overlap therefore the shapes are disjoint.


Edited by Aardvajk, 09 June 2014 - 01:50 AM.





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