MATLAB: Solve an equation for perspective projection

solve equation

Hi guys I got an equation need to simplify but am new to simplify command

D^2=(xa-xb)^2+(ya-yb)^2+(Za-Zb)^2

knowing that:

xa=-f*Xa/Za

xb=-f*Xb/Zb

ya=-f*Ya/Za

yb=-f*Yb/Zb

what i want is to simplify the equation to this form :

af^2+bf+c=0

thank you for help in advance regards,

Best Answer

Hi,

I'm sure this isn't the best solution, but if you enter f, Xa,Xb,Ya,Yb,Za, and Zb as symbols, and then enter your four constraints, and then your equation for D^2, and then use expand, you can pretty easily read off the terms and group them as you like in the quadratic form. I am sure there is a better way, but thought this might help you move along until someone has a better idea.

syms f Xa Xb Za Zb Ya Yb xa=-f*Xa/Za;

xb=-f*Xb/Zb;

ya=-f*Ya/Za;

yb=-f*Yb/Zb;

D2=((xa-xb)^2+(ya-yb)^2+(Za-Zb)^2);

expand(D2)

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MATLAB: Find the volume of the region bounded between the planes x + y + 2z = 2 and 2x + 2y + z = 4 in the first octant.

clear
clc
syms x y z
xa=0;
xb=2;
ya=0+0*x;
yb=2-x;
za=(2-x-y)/2;
zb=4-(2*x)-(2*y);
I=int(int(int(1+0*z,z,za,zb),y,ya,yb),x,xa,xb)
viewSolidone(z,za,zb,y,ya,yb,x,xa,xb)
%Try this code, it will work.

MATLAB: Solving two equations with two unknows

You could undoubtedly solve these two equations using 'solve' of the Symbolic Toolbox or 'fsolve' of the Optimization Toolbox. However, this is the classic problem of finding the two intersections of two circles which is well understood, and the following matlab code is very likely more efficient than either of these methods.

Let P1 and P2 be column vectors containing the coordinates of the two circles' centers - [xA;yA] and [xB;yB] in your case - and let r1 and r2 be the two respective radii. Then the two intersections of the circles are points Q1 and Q2 where Q1 lies off to the left and Q2 off to the right as you move from P1 toward P2.

P21 = P2-P1;
d2 = sum((P21).^2);
P0 = (P1+P2)/2+(r1+r2)*(r1-r2)/2/d2*P21;
Q0 = sqrt(((r1+r2)^2-d2)*(d2-(r1-r2)^2))/2/d2*[0,-1;1,0]*P21;
Q1 = P0+Q0;
Q2 = P0-Q0;

Note: For the above Q0 computation to work properly it is essential that P1 and P2 be column vectors, not row vectors. If they are row vectors, the Q0 code should be changed to its transpose:

Q0 = sqrt(((r1+r2)^2-d2)*(d2-(r1-r2)^2))/2/d2*P21*[0,1;-1,0];

Best Answer

Related Solutions

MATLAB: Find the volume of the region bounded between the planes x + y + 2z = 2 and 2x + 2y + z = 4 in the first octant.

MATLAB: Solving two equations with two unknows

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