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I have an engineering task to solve. It is not a homework assignment. I may not be able to formulate it strictly. I have a 2-sphere positioned in the center of Cartesian Coordinate System with the North Pole on vertical axis Z. The radius is 1.0. I also have a rectangle, unrelated to the sphere. The rectangle is real, it is a screen of a webcam, the sphere is virtual. The screen is filled with pixels. There are 320 pixels horizontally and 240 vertically, that is the ratio R = 1.33… I need to map this rectangle on the Northern hemisphere of the 2-sphere in such a way that the center of the rectangle will coincide with the North Pole of the 2-Sphere and all 4 corners of the rectangle would touch a parallel with the angular distance D from the equator. Using a geographic analogy, the parallel might be the lattitude of Cairo, the Capital of Egypt, or so on. I need a formula of mapping, a mathematical correspondence between two surfaces. If I get such a formula, I can use it in C++ programming in direct calculations with the webcam as input. I would appreciate any help.
Thank you AboutFace 22 ( talk) 15:19, 10 March 2020 (UTC)
Thank you. The way I visualize it, the circle of latitude cannot project onto a circle on the screen. My understanding is that projection of the rectangle on the screen, if linear, will touch the circle of latitude only at four points which are corners of the rectangle, and the sides of the rectangle will lie close to the North Pole, although the elevation might not be significant. Could it be this way?
I need to think about other questions before answering. Thank you AboutFace 22 ( talk) 18:10, 10 March 2020 (UTC)
Second question. Linearity of the projection is very desirable. It is a computer vision system. The stipulation is that the image on the sphere will be expanded in a series of Spherical Harmonics and rotational invariants computed. Nonlinear distortions are very undesirable. AboutFace 22 ( talk) 18:16, 10 March 2020 (UTC)
Gnomonic projections are very interesting. They are almost an answer to my question. Thank you much. As long as they are linear, one of them perhaps could be chosen and implemented. It is nice to have a formula, however, a formula is almost a must. AboutFace 22 ( talk) 18:23, 10 March 2020 (UTC)
Stereographic projection is not acceptable. This is why: "The projection is defined on the entire sphere," I must consider only half of the sphere, because I need to preserve an analogy with a human eye. Thank you, AboutFace 22 ( talk) 18:27, 10 March 2020 (UTC)
Thank you. It is important what you said. Then I have to back off. So, the whole projection is not linear then perhaps locally, around the North Pole it may be approximately linear. This is probably what happens in real human eye. I need to read your mathematical notations and think about them. Thank you, AboutFace 22 ( talk) 20:26, 10 March 2020 (UTC)
If A,B,C,D are four corners of the rectangle (the screen of the webcam) it seems to follow from your explanation that lines AB, BC, CD, DA will not lie on the great circles of the sphere (or they will?) once projection is done, correct? I am reluctant to name the projection, perhaps it should be gnomic? Thank you, - AboutFace 22 ( talk) 20:34, 10 March 2020 (UTC)
Thank you very much. I think I can even use the formula at the Stereographic projection page. It is something to study and implement. I appreciate your help. AboutFace 22 ( talk) 02:07, 11 March 2020 (UTC)
Mathematics desk | ||
---|---|---|
< March 9 | << Feb | March | Apr >> | Current desk > |
Welcome to the Wikipedia Mathematics Reference Desk Archives |
---|
The page you are currently viewing is a transcluded archive page. While you can leave answers for any questions shown below, please ask new questions on one of the current reference desk pages. |
I have an engineering task to solve. It is not a homework assignment. I may not be able to formulate it strictly. I have a 2-sphere positioned in the center of Cartesian Coordinate System with the North Pole on vertical axis Z. The radius is 1.0. I also have a rectangle, unrelated to the sphere. The rectangle is real, it is a screen of a webcam, the sphere is virtual. The screen is filled with pixels. There are 320 pixels horizontally and 240 vertically, that is the ratio R = 1.33… I need to map this rectangle on the Northern hemisphere of the 2-sphere in such a way that the center of the rectangle will coincide with the North Pole of the 2-Sphere and all 4 corners of the rectangle would touch a parallel with the angular distance D from the equator. Using a geographic analogy, the parallel might be the lattitude of Cairo, the Capital of Egypt, or so on. I need a formula of mapping, a mathematical correspondence between two surfaces. If I get such a formula, I can use it in C++ programming in direct calculations with the webcam as input. I would appreciate any help.
Thank you AboutFace 22 ( talk) 15:19, 10 March 2020 (UTC)
Thank you. The way I visualize it, the circle of latitude cannot project onto a circle on the screen. My understanding is that projection of the rectangle on the screen, if linear, will touch the circle of latitude only at four points which are corners of the rectangle, and the sides of the rectangle will lie close to the North Pole, although the elevation might not be significant. Could it be this way?
I need to think about other questions before answering. Thank you AboutFace 22 ( talk) 18:10, 10 March 2020 (UTC)
Second question. Linearity of the projection is very desirable. It is a computer vision system. The stipulation is that the image on the sphere will be expanded in a series of Spherical Harmonics and rotational invariants computed. Nonlinear distortions are very undesirable. AboutFace 22 ( talk) 18:16, 10 March 2020 (UTC)
Gnomonic projections are very interesting. They are almost an answer to my question. Thank you much. As long as they are linear, one of them perhaps could be chosen and implemented. It is nice to have a formula, however, a formula is almost a must. AboutFace 22 ( talk) 18:23, 10 March 2020 (UTC)
Stereographic projection is not acceptable. This is why: "The projection is defined on the entire sphere," I must consider only half of the sphere, because I need to preserve an analogy with a human eye. Thank you, AboutFace 22 ( talk) 18:27, 10 March 2020 (UTC)
Thank you. It is important what you said. Then I have to back off. So, the whole projection is not linear then perhaps locally, around the North Pole it may be approximately linear. This is probably what happens in real human eye. I need to read your mathematical notations and think about them. Thank you, AboutFace 22 ( talk) 20:26, 10 March 2020 (UTC)
If A,B,C,D are four corners of the rectangle (the screen of the webcam) it seems to follow from your explanation that lines AB, BC, CD, DA will not lie on the great circles of the sphere (or they will?) once projection is done, correct? I am reluctant to name the projection, perhaps it should be gnomic? Thank you, - AboutFace 22 ( talk) 20:34, 10 March 2020 (UTC)
Thank you very much. I think I can even use the formula at the Stereographic projection page. It is something to study and implement. I appreciate your help. AboutFace 22 ( talk) 02:07, 11 March 2020 (UTC)