Euler
Examplesby R. Grothmann
This notebook demonstrates analytic geometry in Euler, numerically in the Euler matrix language, and in symbolic form using Maxima.
The geometry functions are in an Euler file geometry.e.
>load geometry
Numerical and symbolic geometry.
We set the range for our plot. The plots of geometry objects all hold the old plot and add the objects to it.
>setPlotRange(-0.5,2.5,-0.5,2.5);
Now set three points and plot them.
>A=[1,0]; plotPoint(A,"A"); >B=[0,1]; plotPoint(B,"B"); >C=[2,2]; plotPoint(C,"C");
Then three segments.
>plotSegment(A,B,"c"); >plotSegment(B,C,"a"); >plotSegment(C,A,"b");
The geometry functions include functions to create lines and circles. The format for the line is [a,b,c], which represents the line with equation ax+by=c.
>lineThrough(B,C)
[-1, 2, 2]
Compute the perpendicular line through A on BC.
>ha=perpendicular(A,lineThrough(B,C));
And its intersection with BC.
>H=lineIntersection(ha,lineThrough(B,C));
Plot that.
>plotPoint(H,value=1); >plotSegment(A,H):
Compute the area of ABC.
>norm(A-H)*norm(B-C)/2
1.5
Compare with determinant formula.
>areaTriangle(A,B,C)
1.5
The length of the height.
>distance(A,H)
1.3416407865
The angle at C.
>degprint(computeAngle(B,C,A))
36°52'11.63''
Now the circumcircle of the triangle.
>c=circleThrough(A,B,C); getCircleRadius(c)
1.17851130198
>plotPoint(getCircleCenter(c),"U"); >plotCircle(c,"Circumcircle"):
This the midpoint of the circle.
>fracformat; getCircleCenter(c), longformat;
[7/6, 7/6]
Now we compute the intersection of two angle bisectors, which is the center of the inscribed circle.
>l=angleBisector(A,C,B); g=angleBisector(C,A,B); >IP=lineIntersection(l,g)
[0.860379610028, 0.860379610028]
Add everything to the plot.
>color(5); plotLine(l); plotLine(g); color(1); >plotPoint(IP,"I"); >plotCircle(circleWithCenter(IP,norm(IP-projectToLine(IP,lineThrough(A,B))))):
We can compute exact and symbolic geometry using Maxima.
The file geometry.e provides the same (and more) functions in Maxima. However, we can use symbolic computations now.
>A &= [1,0]; B &= [0,1]; C &= [2,2];
The functions for lines and circle work just like the Euler functions, but provide symbolic computations.
>c &= lineThrough(B,C)
[- 1, 2, 2]
We can get the equation for a line easily.
>&getLineEquation(c,x,y), &solve(%,y) | expand
2 y - x = 2
x
[y = - + 1]
2
>&getLineEquation(lineThrough(A,[x1,y1]),x,y)
(x1 - 1) y - x y1 = - y1
>ha &= perpendicular(A,lineThrough(B,C))
[2, 1, 2]
>H &= lineIntersection(c,ha)
2 6
[-, -]
5 5
>&projectToLine(A,lineThrough(B,C))
2 6
[-, -]
5 5
>&distance(A,H)
3
-------
sqrt(5)
>cc &= circleThrough(A,B,C)
7 7 5
[-, -, ---------]
6 6 3 sqrt(2)
>&getCircleRadius(cc)|float
1.178511301977579
>&computeAngle(A,C,B)
4
acos(-)
5
>&solve(getLineEquation(angleBisector(A,C,B),x,y),y)[1]
y = x
>IP &= lineIntersection(angleBisector(A,C,B),angleBisector(C,B,A))
sqrt(2) sqrt(5) + 2 sqrt(2) sqrt(5) + 2
[-------------------, -------------------]
6 6
>IP()
[0.860379610028, 0.860379610028]
Of course, we can also intersect lines with circles, and circles with circles.
>A &:= [1,0]; c=circleWithCenter(A,4); >B &:= [1,2]; C &:= [2,1]; l=lineThrough(B,C); >setPlotRange(5); plotCircle(c); plotLine(l);
The intersection of a line with a circle returns two points and the number of intersection points.
>{P1,P2,f}=lineCircleIntersections(l,c);
>P1, P2,
[4.64575131106, -1.64575131106] [-0.645751311065, 3.64575131106]
>plotPoint(P1); plotPoint(P2):
The same in Maxima.
>c &= circleWithCenter(A,4)
[1, 0, 4]
>l &= lineThrough(B,C)
[1, 1, 3]
>&lineCircleIntersections(l,c) | radcan,
[[sqrt(7) + 2, 1 - sqrt(7)], [2 - sqrt(7), sqrt(7) + 1]]
Now we check the chord angle theorem.
>C=A+normalize([-2,-3])*4; plotPoint(C); plotSegment(P1,C); plotSegment(P2,C); >computeAngle(P1,C,P2)
1.20942920289
>C=A+normalize([-4,-3])*4; plotPoint(C); plotSegment(P1,C); plotSegment(P2,C); >computeAngle(P1,C,P2)
1.20942920289
>insimg;
Let us construct the middle perpendicular the usual way.
>A=[2,2]; B=[-1,-2];
>c1=circleWithCenter(A,distance(A,B));
>c2=circleWithCenter(B,distance(A,B));
>{P1,P2,f}=circleCircleIntersections(c1,c2);
>l=lineThrough(P1,P2);
>setPlotRange(5); plotCircle(c1); plotCircle(c2);
>plotPoint(A); plotPoint(B); plotLine(l):
Next, we do the same in Maxima with general coordinates.
>A &= [a1,a2]; B &= [b1,b2]; >c1 &= circleWithCenter(A,distance(A,B)); >c2 &= circleWithCenter(B,distance(A,B)); >is &= circleCircleIntersections(c1,c2); P1 &= is[1]; P2 &= is[2];
The equations for the intersections are quite involved. But we can simplify, if we solve for y.
>&solve(getLineEquation(lineThrough(P1,P2),x,y),y)
2 2 2 2
- (2 b1 - 2 a1) x + b2 + b1 - a2 - a1
[y = -----------------------------------------]
2 b2 - 2 a2
This is indeed the same as the middle perpendicular, which is computed in a completely different way.
>&solve(getLineEquation(middlePerpendicular(A,B),x,y),y)
2 2 2 2
- (2 b1 - 2 a1) x + b2 + b1 - a2 - a1
[y = -----------------------------------------]
2 b2 - 2 a2