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资源说明：Parametric Curve Manipulation
-- FIRST ASSIGNMENT -- PARAMETRIC CURVES --
Name: Wagner Schmitt
UFRGS: 159824
1. BUILD:
The program binary was built in Ubuntu x64. To build it again the user needs to browse the main directory where the makefile is located, and type make.
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2. USING THE PROGRAM: CONDITIONS
As soon as the program starts to run, the user will be presented with a black screen with a floating menu. This menu is where the user can input his own basis aside from define some other settings.
2.1. == <** USER BASIS **>
The program starts with a default basis defining a Hermite Curve. After the user change any configuration in the menu, HE SHOULD CLICK [[UPDATE CHANGES]] TO APPLY THE CHANGES.
If the basis defined by the user wasn't define a basis (the determinant of the basis matrix equals to zero), no curve will be drawn, and the menu field 'determinant' will be displayed zero (determinant = 0).
2.1. == <** POINTS **>
Next, the user should provide four points p1,p2,p3 and p4, by clicking in diferents position on the screen. As soon he provide the points, a curve will be drawed. After that point, the user can interact with the control points or the curve itself.
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3. BASIS DERIVATION:
The user can build its own basis provinding values for P1,P2,P3,P4, in [0,1]. Also, he defines witch is a POINT or a TANGENT.
Assuming the user specified the four points P1,P2,P3,P4:
P1(t1) = a(t1)^3 + b(t1)^2 + c(t1) + d;
P2(t2) = a(t2)^3 + b(t2)^2 + c(t2) + d;
P3(t3) = a(t3)^3 + b(t3)^2 + c(t3) + d;
P4(t4) = a(t4)^3 + b(t4)^2 + c(t4) + d;
So its computed:
[ P1 ] [ (t1)^3 (t1)^2 (t1) 1 ] [ a ]
[ P2 ] = [ (t2)^3 (t2)^2 (t2) 1 ] x [ b ]
[ P3 ] [ (t3)^3 (t3)^2 (t3) 1 ] [ c ]
[ P4 ] [ (t4)^3 (t4)^2 (t4) 1 ] [ d ]
To find the coeficients [a b c d ]^T, the basis matrix is inverted:
[ a ] [ (t1)^3 (t1)^2 (t1) 1 ] (-1) [ P1 ]
[ b ] = [ (t2)^3 (t2)^2 (t2) 1 ] X [ P2 ]
[ c ] [ (t3)^3 (t3)^2 (t3) 1 ] [ P3 ]
[ d ] [ (t4)^3 (t4)^2 (t4) 1 ] [ P4 ]
So the Basis function is :
[ (t1)^3 (t1)^2 (t1) 1 ] (-1) [ P1x P1y ]
Q(t) = [ t^3 t^2 t 1 ] x [ (t2)^3 (t2)^2 (t2) 1 ] X [ P2x P2y ]
[ (t3)^3 (t3)^2 (t3) 1 ] [ P3x P3y ]
[ (t4)^3 (t4)^2 (t4) 1 ] [ P4x P4y ]
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4. MINIMUM DISTANCE FROM POINT TO THE PARAMETRIC CURVE
Assuming a parametric curve Q(t) = ( X(t) Y(t) )
the curve will produce the points P( X(t), Y(t) ) from t varying in [0,1].
So the distance from an arbitrary point P( Px, Py ) to any point P( X(t), Y(t) ) is given by:
D(t)^2 = (X(t)-Px)^2 + (Y(t)-Py)^2
Since, D is in function of 't', we want to find the values of t wich minimizes D.
Minimize D(t)^2 = (X(t)-Px)^2 + (Y(t)-Py)^2
Such as: t in [0,1]
To solve this we have to find the critical points where D'(t) = 0.
and test wich of the values of t gives the minimum distance in D(t).
Obs: This will produce a polynomium of degree 5, so it will have 5 critical points.
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5. CODE FROM OTHER SOURCES
5.1 - JERKINS-TRAUB NUMERICAL ALGORITHM TO FIND ROOTS OF POLINOMIUM OF ANY DEGREE:
It was used the method of Jenkins-Traub to find the roots of the polynomium of 5 dregree. The algoritm was modified to fulfill the needs of the program.
Source Code: http://www.akiti.ca/rpoly_ak1_Intro.html
5.2 - ALGORITHM FOR COMPUTING THE DETERMINANT AND INVERSE OF A MATRIX:
Written by Mike Dinolfo.
Source Code: users.erols.com/mdinolfo/matrix.htm
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