# Common Tangent to two Ellipses

Author:
Thijs
Here's a elegant solution, if I do say so myself. ;-) The beauty is in the function: d(α)= d1*cos(2α) + d2*sin(2α) - d3*cos(α) - d4*sin(α) + d5 I don't have any intuitive sense of the geometric reason for this, but it works. Setup #======================================== # Common Tangent to two Ellipses #======================================== E1a = (-4, 1) E1b = (-2,-1) E1c = (-5,-1) ellipse1 = Ellipse(E1a, E1b, E1c) E2a = ( 3, 1) E2b = ( 2,-1) E2c = ( 4, 0) ellipse2 = Ellipse(E2a, E2b, E2c) # Transform the coordinate system (without loss of generality) # so that ellipse1 becomes a standard circle (x^2 + y^2 = 1) #---------------------------------------------------------------------- Cen1 = Center(ellipse1) Maj1 = MajorAxis(ellipse1) rot = atan2d(x(Maj1), y(Maj1)) + 180° a1 = SemiMajorAxisLength(ellipse1) b1 = SemiMinorAxisLength(ellipse1) Mt = {{1,0,-x(Cen1)}, {0,1,-y(Cen1)}, {0,0,1}} Mr = {{cos(rot),-sin(rot),0}, {sin(rot),cos(rot),0}, {0,0,1}} Ms = {{1/a1,0,0}, {0,1/b1,0}, {0,0,1}} Me = Ms Mr Mt # Create matrix representation of ellipse2 (line-conic) #---------------------------------------------------------------------- e2 = ApplyMatrix(Me, ellipse2) c = Coefficients(e2) M2 = {{c(1),c(4)/2,c(5)/2},{c(4)/2,c(2),c(6)/2},{c(5)/2,c(6)/2,c(3)}} iM2= Invert(M2) # Make a function for "signed distance" between tangent and e2 # If distance = 0 then tangent of circle is tangent of ellipse e2 #---------------------------------------------------------------------- d1 = iM2(1,1) - iM2(2,2) d2 = 2*iM2(1,2) d3 = 4*iM2(1,3) d4 = 4*iM2(2,3) d5 = iM2(1,1) + iM2(2,2) + 2*iM2(3,3) d(α)= d1*cos(2α) + d2*sin(2α) - d3*cos(α) - d4*sin(α) + d5 Sn = Solutions(d = 0) # Transform tangents back to the original coordinate system #---------------------------------------------------------------------- iMe = Invert(Me) t1 = ApplyMatrix(iMe,(cos(Sn(1))*x + sin(Sn(1))*y = 1)) t2 = ApplyMatrix(iMe,(cos(Sn(2))*x + sin(Sn(2))*y = 1)) t3 = ApplyMatrix(iMe,(cos(Sn(3))*x + sin(Sn(3))*y = 1)) t4 = ApplyMatrix(iMe,(cos(Sn(4))*x + sin(Sn(4))*y = 1)) # Some setting for created objects #---------------------------------------------------------------------- Lobj = {"E1a", "E1b", "E1c", "ellipse1"} Execute(Zip("SetDynamicColor("+obj+",1,0,0)", obj,Lobj)) Execute(Zip("ShowLabel("+obj+",false)", obj,Lobj)) Lobj = {"E2a", "E2b", "E2c", "ellipse2"} Execute(Zip("SetDynamicColor("+obj+",0,0,1)", obj,Lobj)) Execute(Zip("ShowLabel("+obj+",false)", obj,Lobj)) Lobj = {"Cen1", "Maj1", "e2", "d"} Execute(Zip("SetVisibleInView("+obj+",1,false)", obj,Lobj)) Lobj = {"t1", "t2", "t3", "t4"} Execute(Zip("ShowLabel("+obj+",false)", obj,Lobj)) Delete(Lobj)