Solving system of equations using approximation

asked 2014-08-29 18:58:46 +0100

CCorio
3 ●1

Hello - I'm an experienced software developer but new to Sage. I'm trying to solve a system of equations in various ways using Sage online.

Here's the system of equations

var('x y h k r t d')

eq1 = x^2 + y^2 == d^2

eq2 = y == tan(t) * x

eq3 = (x-h)^2 + (y-k)^2 == r^2

eq4 = -1 * r < h < r

eq5 = -1 * r < k < r

I'm trying to solve this in two ways:

I'm trying to solve for h and k for a given r using a set of t and d values. This will obviously be an approximation.
I'm trying to get all values of d given h, k, r, and t. I think there should be two values. I was trying to get the equations for d but that might not be possible...

If you could provide any insight into the functions to use to get these results or any sample code, I'd appreciate it.

Thanks, Chris

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answered 2014-08-29 19:18:27 +0100

FrédéricC

5236 ●3 ●44 ●116

Well, as this is a purely polynomial problem (by replacing $\tan(t$) by $t$) of intersection between two circles, you can use polynomial tools.

sage: x,y,d,t,h,k,r = polygen(QQ,'x,y,d,t,h,k,r')
sage: eq = [x*x+y*y-d*d,y-t*x,(x-h)**2+(y-k)**2-r**2,h-1,k-1,r-6,t-2]
sage: ring = x.parent()
sage: ideal = ring.ideal(eq)
sage: ideal.dimension()
0
sage: ideal.variety()   # no solutions with rational coordinates
[]
sage: ideal.variety?   # help of the variety method
sage: ideal.variety(ring=CC)     # working over the complex numbers
[{y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}]
sage: ideal.variety(ring=RR)     # working over the real numbers
[{y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: -4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: -4.15163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 4.64166933416076, k: 1.00000000000000, h: 1.00000000000000, x: -2.07581763205193}, {y: 6.55163526410386, r: 6.00000000000000, t: 2.00000000000000, d: 7.32495090716051, k: 1.00000000000000, h: 1.00000000000000, x: 3.27581763205193}]

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can you help me understand this equation set a bit more? eq = [x*x+y*y-d*d,y-t*x,(x-h)**2+(y-k)**2-r**2,h-1,k-1,r-6,t-2] What does the "h-1,k-1,r-6,t-2" signify? Why is it -1 or -6, etc.?

CCorio ( 2014-08-29 19:31:49 +0100 )edit

The last 4 equations in eq are there to set the values of h,k,r and t. I have chosen arbitrary values.

FrédéricC ( 2014-08-29 19:53:15 +0100 )edit

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