Smart Phone Panoramas

Geometry Level 5

Many smart phones today are able to take panorama shots, by sweeping the camera from one direction to another.\

Assuming ideal pinhole optics and that the camera works like a slit scan, all straight lines of the object become what family of curves?

Assume that, mathematically speaking, all points of the object are projected onto the surface of a cylinder, by means of rays that all pass through a common point on the axis of the cylinder. Then the surface of the cylinder is unrolled flat to create the final image.

1 Conic curves in general, depending on orientation of straight lines in object

2 Parabolic curves only

3 Circular and elliptical curves only

4 Hyperbolic curves only

5 Trigonometric curves in general, depending on orientation of straight lines in object

6 Sinusoidal curves only (or Cosinusoidal if you prefer--they differ only by phase)

7 Cosecant curves only (or Secant if you prefer--they differ only by phase)

8 Not limited to conic or trigonometric curves

If either 2, 3, or 4 is true, do not choose 1. If either 6 or 7 is true, do not choose 5.

5 3 8 4 1 2 7 6

1 solution

Michael Mendrin
Aug 16, 2015

Any straight line in the object and the common point which all rays pass through define a plane which intersects the cylinder. The function of the curve on the surface of the cylinder is then of the form

$aSin\left( \theta \right) +bCos\left( \theta \right) =z$

but this is the same as

$\sqrt { { a }^{ 2 }+{ b }^{ 2 } } Sin\left( \theta +ArcTan\left( \dfrac { b }{ a } \right) \right) =z$

Hence, all straight lines in the object become sinusoidal curves only.

Note: Many real panorama cameras attempt to make a wide angle shot vertically while doing a slit scan (especially the 360 degree cameras), so there is distortion towards top and bottom of the pictures, resulting in flattening of the peaks of the sine curves, i.e., it's not a true cylindrical projection.

Example of smart phone panorama

This one was shot inside a Japanese market, given to me by a friend. I got the idea of this problem from this picture.

Michael Mendrin - 5 years, 10 months ago

When I take panorama I noticed the problem like you: the image is being bent. That's why I take many photos and then merge it.

Adam Phúc Nguyễn - 5 years, 10 months ago

If one "takes many photos" in successive changes in direction, and then merge them, the result will be the same. it's still basically a cylindrical projection. It's similar to the problem of trying to make a flat map of the world. In fact, if the world is the object, then the cylindrical projection of it would be the Mercator Map. But you don't see any sine waves in it because there aren't any straight lines on the globe of Earth.

By the way, congrats for solving this problem.

Michael Mendrin - 5 years, 10 months ago

Smart Phone Panoramas

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