*Parametric CAD*

¥
__Parametric Computer Aided Geometric Design, __

__ Automatic Geometric Reasoning__

Hoffman, Kapur, Chou, Wu, Wang, Buchberger, Joan-Arinyo

General Electric

Finding solutions to an initial sketch with restrictions

(distance, tangency, perpendicularity, incidence, collinearity, etc..)

distance(a,b)=d1

distance(b,c)=d2

distance(a,d)=d3

tangent(segment (a,b), circle(Q))

tangent(segment (b,c), circle(Q))

on(d,circle(Q))

on(c,circle(Q))

Vision, Pattern recognition

-consistency: given two images, to determine if they come from one

conjetural scene

-obtaining restrictions to the scene as imposed by image properties.

__Hypotheses or Construction__
:

¥polynomial equations,

¥inequations

eg. relation between given and projected points, sketch...

__Thesis or Conclusion__
:

¥eg. paralellism,

¥existence of solution,

¥perpendicularity condition, etc..

Symbolic coefficients: integers, integers+parameters

¥Checking if thesis follows from hypotheses:

__Automatic proving__

¥Searching for complementary hypotheses for the thesis to become true

AUTOMATIC DISCOVERY

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__Example__
(with CoCoA):

For a given triangle, we claim the circumcenter lies on one side.

(x,y) circumcenter

Theorem is false:

*NormalForm(1,Ideal(x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, yt-1))
1;*

Theorem does not provide degeneracy complementary hypotheses:

Ideal( 0 );

Theorem provides complementary restrictions

Ideal( a^2e + b^2e - ae^2 );

We add this new hypothesis and look for degeneracy conditions in the new situation:

*
Ideal( a^2e - ae^2 ,
be );*

Hence, if result holds then:

be0 or a^2e - ae^20

In particular,

*be0, a^2+b^2-ae=0,*

ie. rectangular triangle.

*NormalForm(1,Ideal(x^2+y^2-(x-e)^2-y^2,x^2+y^2-(x-a)^2-(y-b)^2, a^2e+b^2e-ae^2, beh-1, yt-1))
0*