Literally here. The problem goes this way. Looking to find the plane intercept of two intersecting cylinders whose intersection isn't perpendicular. The intersection in this case is like the case of conics if you are curious, albeit this is a cylindrical elliptic as opposed to generated from in an increasing or decreasing radial dimension on the surface of the conic where the plane slices at some angle other then 90 or 0 degrees. In this case, while one might be tempted to resort to something like cylindrical geometry where resources of this sort exist...more commonly you'd likely find spherical trigonometry resources more quickly. The simplest solution relies upon using projective geometry. If we project the image of the cylinder in two dimensions where the intercept is to occur, then we can apply at a given angle more familiar trigonometric rules. The chosen projected image of the ellipse on the local 'yz' in our next example is such that image of local x domain is twice mapped into a resulting right triangle for the given projection
...implying that familiar right triangle trigonometric rules should apply. Here we choose local coordinate systems such that the chord length a is on local y, and the chord length b is chosen on the local x. I'd leave it up to you to find the conversion from global to local coordinates here on this problem. We'd know automatically that chord length b in the cylinder conic is simply the radius of the cylinder...why is this, in the projective since since local z in the projective mapping on the chord length is the same, implies the conversion in the projective sense makes local z irrelevant for computation of distance ( magnitude) on the local chord length b axis, thus magnitude in preserved from with local x coordinate conversions. For example,
Where the intersection is projected on the local 'yz' plane the following identity:
tan phi = z / yc where yc is the unique coordinate system for our given projection and phi is in the intersection angle.
noting that any x, y position on the cylinder whose height (length) axis runs tangent to the z, means that any x or y position is = < radius of the cylinder. The same is true for any plane intercept for on such cylinder at a given angle theta. Thus translating coordinate systems to yc, a simple coordinate change yc = radius +/- y depending on the orientation of the intercept.
Here it seems then one can apply the more familiar top down projection on such cylinder to find the coordinates of y at a point, or in other words, polar coordinate system we could compute y as follows. Or if working in units of arc radians on your given subdivision of the common circle this could look like:
y = cylinder radius * sin theta where theta represent some arc subdivision of 2 pi radians or 360 degrees.
similarly we can find x using:
x = cylinder radius * cos theta
and z is found once having computed y. This actually generates the elliptical on the intersecting plane likewise.
You could also find the chord lengths a, and find the elliptic equation computing local y which is theta = {0, 90}, which results in a distance magnitude of cylinder radius. We compute the distance zmag computing from local z inputting and converting from yc values, then apply the right triangle formula:
a^2 = r^2 + zmag^2
this provides the chord length of a.
hmm...
very informal thoughts no formal proofs here....problem relates to projective geometry. Say a circle exists on the local xy plane, utilizing the right projection one can project this to a line where choosing a projection such that all but two points, x1 and x2 map have a domain doubly mapped into such projective space to the same coincident points in x' (this means x' = x). The problem here is that projecting another circle whom has more then two points neither doubly mapped implies that use of right triangle formulation breaks down, since invariably one circle neither fully aligned to the projective axis will instead by described as an arc on such projective plane, and neither a line?! I've thought of this alternately, and as it turns out I believe one can still apply the methods above with respect to finding the formulation for the ellipse by way of chord lengths. Here, choosing the local projective axis such that all but two points x1, and x2 on the ellipse are doubly mapped and are represented as a line of chord length a for the projective plane. Since in such projective plane where axis are represented by coordinates y' and z' we know that preservation of magnitude should exist in so far as the cylinders local y, this implies that the y' length should be 2 cylinder radius since both such y1 and y2 have no other possible coordinate representation in y' for the respective coordinate companions x1 and x2 . Which means one can use standard trigonometric formulation here to find the chord length b,
2a = length(y1, y2) * (sin phi)^-1
I believe similar argument hold for chord length a = cylinder radius with our given intersection...note this is not a skew intersection but where both such cylinder center line (local yz) intersect.
Phi then determines the angle of coordinate conversion and the formula for the ellipse is given by
x ' ^2 / a + y'^2 / b = 1
Then would just need to convert to local x,y,z...here x' = x and z' = 0.
Reason for this...actually motivated in the applied sense. Trying to mathematically describe hand rails on stairs :)
Followup: Put the first mentioned method to test for mesh construction, appears to generate an ellipse or an ellipsoidal curve.
...implying that familiar right triangle trigonometric rules should apply. Here we choose local coordinate systems such that the chord length a is on local y, and the chord length b is chosen on the local x. I'd leave it up to you to find the conversion from global to local coordinates here on this problem. We'd know automatically that chord length b in the cylinder conic is simply the radius of the cylinder...why is this, in the projective since since local z in the projective mapping on the chord length is the same, implies the conversion in the projective sense makes local z irrelevant for computation of distance ( magnitude) on the local chord length b axis, thus magnitude in preserved from with local x coordinate conversions. For example,
Where the intersection is projected on the local 'yz' plane the following identity:
tan phi = z / yc where yc is the unique coordinate system for our given projection and phi is in the intersection angle.
noting that any x, y position on the cylinder whose height (length) axis runs tangent to the z, means that any x or y position is = < radius of the cylinder. The same is true for any plane intercept for on such cylinder at a given angle theta. Thus translating coordinate systems to yc, a simple coordinate change yc = radius +/- y depending on the orientation of the intercept.
Here it seems then one can apply the more familiar top down projection on such cylinder to find the coordinates of y at a point, or in other words, polar coordinate system we could compute y as follows. Or if working in units of arc radians on your given subdivision of the common circle this could look like:
y = cylinder radius * sin theta where theta represent some arc subdivision of 2 pi radians or 360 degrees.
similarly we can find x using:
x = cylinder radius * cos theta
and z is found once having computed y. This actually generates the elliptical on the intersecting plane likewise.
You could also find the chord lengths a, and find the elliptic equation computing local y which is theta = {0, 90}, which results in a distance magnitude of cylinder radius. We compute the distance zmag computing from local z inputting and converting from yc values, then apply the right triangle formula:
a^2 = r^2 + zmag^2
this provides the chord length of a.
hmm...
very informal thoughts no formal proofs here....problem relates to projective geometry. Say a circle exists on the local xy plane, utilizing the right projection one can project this to a line where choosing a projection such that all but two points, x1 and x2 map have a domain doubly mapped into such projective space to the same coincident points in x' (this means x' = x). The problem here is that projecting another circle whom has more then two points neither doubly mapped implies that use of right triangle formulation breaks down, since invariably one circle neither fully aligned to the projective axis will instead by described as an arc on such projective plane, and neither a line?! I've thought of this alternately, and as it turns out I believe one can still apply the methods above with respect to finding the formulation for the ellipse by way of chord lengths. Here, choosing the local projective axis such that all but two points x1, and x2 on the ellipse are doubly mapped and are represented as a line of chord length a for the projective plane. Since in such projective plane where axis are represented by coordinates y' and z' we know that preservation of magnitude should exist in so far as the cylinders local y, this implies that the y' length should be 2 cylinder radius since both such y1 and y2 have no other possible coordinate representation in y' for the respective coordinate companions x1 and x2 . Which means one can use standard trigonometric formulation here to find the chord length b,
2a = length(y1, y2) * (sin phi)^-1
I believe similar argument hold for chord length a = cylinder radius with our given intersection...note this is not a skew intersection but where both such cylinder center line (local yz) intersect.
Phi then determines the angle of coordinate conversion and the formula for the ellipse is given by
x ' ^2 / a + y'^2 / b = 1
Then would just need to convert to local x,y,z...here x' = x and z' = 0.
Reason for this...actually motivated in the applied sense. Trying to mathematically describe hand rails on stairs :)
Followup: Put the first mentioned method to test for mesh construction, appears to generate an ellipse or an ellipsoidal curve.
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