Can a true right angle be found on a curved surface?

Question

If not, I propose that at the center of a hoop (not in the hoop itself) of light there must be an area that every emanation of light would "miss". Infinity at its smallest existance, perhaps, yet a "lightless" location. (This also assumes a single point of reference touching the hoop. Not three points making a "T" to obtain the right angle "off surface".) This question is for serious physicists.

Answer 1

Sure it can. Assume a curved surface W(x,y,z) and you want a 'right angle' to a given line that passes through point (x0,y0) on the surface.

Compute G = gradient(W) at the point (x0,y0) (note that G is a vector). Now locate three points on a tangent plane (P) by calculating three non-colinear vectors normal to G. Projectively map (parallel to G) all of the points on the line (on W) onto the tangent plane to form a line in the tangent plane. Now draw a line on the tangent plane (or in the subspace spanned by the tangent plane if you think in those terms) and use the inverse of the original projective mapping to map it back into W. That will give you a right angle between the two lines where they intersect at (x0,y0)

Note that, in general, it won't have the same geometric characteristics that a 'right angle' on a plane surface would have since W is the basis space for a non-linear (non-Euclidian) geometry.

It's a bunch of vector calculus, but it's pretty straightforward.


Doug

Answer 2

isn't that sensible?! What an spectacular theory. it would could be quite a brilliant circle. probable this substitute right into a concept test. The ancients have been very enthusiastic relating to the form of the planet earth. Eclipses extremely stricken them quite ones of the moon which confirmed a around development led to by applying the earth reducing off the easy from the solar. instinct informed them that the earth is flat, yet their medical minds mentioned there have been particular data that contradicted this theory. subsequently Aristotle's concept test which you mentioned. The rope is tied on the backside of the pole. the thank you to analyze his test is to attraction to a circle around a ball. positioned a pin the place the middle of the circle is and positioned particularly thread from the pin to the circle you purely drew. If the earth substitute right into a flat floor the circumfence of the rope could be C = 2*pi*r If it substitute into around like your ball, then a number of the radius could be eaten up going from the pin on your circle because of the fact the exterior is around. you will possibly get a style that selection into decrease than what you will get once you do your 2*pi*r calculation. i'm hoping that's obvious. It took me somewhat to comprehend it. in case you decide on added clarification, in basic terms e mail me.

Answer 3

Yes. Consider the curved surface of the Earth. The prime meridian (or any meridian really) and the equator cross at exactly a right angle.