@vv What clever reasoning; the minimum x after any rotation and translation of the ellipse in the corner. This reasoning should apply to any other ellipse-like shape in a corner.

@vv I understand your parametric expressions for the coordinates of the sliding ellipse (Wikipedia helps) but

I must be thick as I've tried and failed to derive the coordinates of xc and yc. Please explain how their expressions arise (a diagram of the ellipse rotated at a sample angle would help).

@vv Does the expression you plot for the sliding ellipse presume that its centre moves in a circle whose centre is the origin?

@Markiyan Hirnyk I forgot to make clear that I have no clue how to prove the path of the sliding ellipse's centre is an arc of a circle.

The first worksheet animates the ellipse sliding against stationary axes , the second animates the axes sliding against the stationary ellipse. The first worksheet uses the technique in the second worksheet. Both are written in Maple15.

MathMech_ellipse_sliding_in_first_quadrant_pg_72.mw

MathMech_tangents_sliding_around_ellipse.mw

Page 72 refers to the book "The Mathematical Mechanic" by Mark Levi, Princeton University Press. This is where I first learned of this problem.

@vv The Alexander horned sphere is one of the wierdest objects I've encountered!

@Kitonum I apologize for not previously thanking you for your correct answer to my question.

It seems that Maple15 cannot display a truly solid object. For example, when I display a hemisphere and ask that the region between it and the xy plane be filled, the projection of the hemisphere on the plane is coloured but the interior of the figure is hollow, as evidenced if you bore a hole through it. Am I correct in this?. Is there a way to display an enclosed 3D figure, such as a sphere, with a fully coloured interior?

@Kitonum

The plottools[transform] command below successfully cuts two smaller spheres of radius CutSphereRadius out of the larger solid sphere of radius one, centre the origin. The smaller spheres are centred at 0.1 and -0.1.

This command shows that plottools[transform] accepts nested `if` commands, thus applying multiple conditions to the transformed plot object.

 transform(proc (x, y, z) options operator, arrow; `if`(.89 < z or z < (-1)*.89, [undefined, undefined, undefined], `if`(sqrt(x^2+y^2+(z-.5)^2) <= CutSphereRadius and (-1)*.1 <= z, [CutSphereRadius*cos(arctan(y, x)), CutSphereRadius*sin(arctan(y, x)), z], `if`(sqrt(x^2+y^2+(z+.5)^2) <= CutSphereRadius and z <= .1, [CutSphereRadius*cos(arctan(y, x)), CutSphereRadius*sin(arctan(y, x)), z], [x, y, z]))) end proc)

A_solid_sphere_with_various_cutouts.mw  @acer I hope I have uploaded correctly and you can access my worksheet.

Thanks for your help now and in the past.

@Christopher2222 Thanks for your answer. I have a Dell Inspiron 1545 with an Intel Pentium Dual Core T4500 at 2.3 GHz, 800 MHz, 1 MB Cache.

Memory is 4 GB DDR2 800 MHz 2 Dimm

Graphics is an Intel Graphics Media Accelerator X4500D

For the past few months the operating system has been Windows 10.

@Carl Love Thanks Carl. Your advice would be useful in specific cases.

@Rouben Rostamian  Your tutorial gave me a good understanding of barycentric coordinates. As evidence of this I modified, in your answer to my question, a copy of your code which maps the planar triangle to the unit circle to map this same triangle to an ellipsoid. I don't see how this could have been done without barycentric coordinates.

A final question; can barycentric coordinates apply to any planar polygon of more than three sides?

@Rouben Rostamian  Rouben, your worksheet shows me the solution to my questions 1 and 2 and I overlooked a simple answer to question 3.

I regularly scan Maple Primes to find new Maple programming techniques and, occasionally, new math techniques and your worksheet has amazed me in both regards.

I had no knowledge of barycentric coordinates but now I see they have great power in solving otherwise tough math problems and they define a vector (V) which provides all the coordinates within the triangle as lambda1 and lambda2 cycle through their ranges.

Try as I might, there are several statements in your worksheet that baffle me.

I can't find a clear explanation or example in Maple 15 help pages of the syntax %T.

Please explain why the correct ranges for plotting the planar triangle are the ones you specified in your plot3d command.

Why does the statement V / sqrt(V^%T . V); not evaluate i.e. acts as an inert command?

Please also explain why plotting the unnormalized V gives a planar triangle while plotting the normalized V plots the same triangle fitted to the surface of the unit sphere.

Please explain why the ranges of function U within the spacecurves of the normalized V provide the edges of the spherical triangle.

Your worksheet is an education in itself and I greatly appreciate your attention.

@vv I moved my archive to my personal documents library and now can successfully save and delete expressions in it. Thank you for your wonderful advice.

BTW the procedure I most recently saved contained the short version of VectorCalculus[CrossProduct] namely &x. When I executed this procedure today &x did not work but replacing it with the full command did work. Yet &x worked in the same procedure yesterday.. a mystery!!

@dharr Yes, these also do the job. Thanks.

@tomleslie Thanks for your help. The warning does look like a bug.