Projection systems are used to convert a 3D shape to a planar (2D) shape.

According to the type of projection system, different results and shapes like rectangles, pies, ellipses, circles, ... can be produced out of a sphere.

Projection systems can be classified by the characteristics of the result they generate.

To continue, I would like to use a very touchable and common example that we have all seen before, Earth sphere and global wide maps, they are everywhere.

# Suppose your sphere is the earth!

Imagine the earth as your sphere and a planar world map that is created from the spherical shape of the earth. In most of the world maps you see the countries near to the poles are getting much bigger than they are in reality, like Iceland which is 1/14 of Africa continent in reality but the map shows them both as equal. This is because when we are omitting one dimension we loose one characteristic of our shapes.

# Different projection systems and their results

*This is a planar projection which doesn't conserve distance, angles or area. The red circles show the amount of exaggeration that is the product of this projection.*

*Equal-Area, look at Iceland and Africa in this one and compare with above.*

Projection systems can be classified by what they preserve.

- Equal area.
- Equal angle which preserve the shape without distortion (conformal).
- Equal distance.
- ......

Conformal projections preserve the shapes but area will not be preserved (the first above picture) this one is the most famous projection system that is used in many applications. Your sphere is a rectangle here!

### So you cannot say a sphere will be projected to an ellipse always. As mentioned above a sphere can be projected to a rectangle (first shape) or can be an ellipse but with different characteristics (equal angle, distance, shape, area - see the following picture), or you may also project a sphere into a conic and then open the conic so you will have a pie.

Each of the above projection systems can be applied with iterative or direct algorithms that can be found on the internet. I didn't talk about the formula and transformations because you didn't ask. Although I wish you to find this answer useful.

### In perspective projections I say yes only ellipses will be produced out of spheres

Cutting a conic with a horizontal plane creates a circle.

Cutting with an oblique plane creates a bevel which would be an ellipse or a hyperbola depending on the cutting angle, and when this angle inclines to be vertical in will create a parabola (following picture).

Maybe this is obvious but take a look at their equations.

For simplicity I assumed all geometries are origin centered.

### Equations:

Circle: $x^2+y^2=r^2$

Ellipse: $x^2/a^2+y^2/b^2=1$

Hyperbola: $x^2/a^2-y^2/b^2=1$

Parabola: $y^2=4ax$

### Morphology :

An ellipse has two foci obviously. A circle as a special kind of ellipsis has two foci too but they are coincident. A hyperbola however is a y axis mirror of its equal ellipsis and it has two foci too. A parabola has one focus but actually it has two because the second one is at infinity: when the cutting plane inclines to 90 degrees (bearing angle), second focus goes to infinity.

# Conclusion

As you see all are ellipses, however you may name them differently to describe special cases, but if you are going to implement it in a game, you need to assume an ellipse equation and it is enough. I can't tell which one of you guys are right, you or your friend, because both could be right.

1

This should go somewhere in this thread, so adding it here :) Inigo Quilez's analytic sphere projection: https://www.shadertoy.com/view/XdBGzd

– Mikkel Gjoel – 2016-02-02T12:11:49.2002

Assuming a perspective projection, AFAICS the 'boundary' formed by the view-points horizon will be a (truncated) cone and thus most of the projection will be a conic section: https://en.wikipedia.org/wiki/Conic_section. An ellipse is thus a possibility, but not the only one.

– Simon F – 2015-09-15T12:27:54.890Pardon my naivety, isn't an ellipse a conic section? Could a projected sphere ever result in a parabola or hyperbola? – hippietrail – 2015-09-15T12:31:23.007

If you look at the wikipedia diagram, https://en.wikipedia.org/wiki/Conic_section#/media/File:Conic_Sections.svg, and consider the plane onto which you are projecting, you can get anything from an ellipse/circle, through to unbounded parabolas or hyperbolas (and I guess if the plane passes through the eye, even degenerate cases)

– Simon F – 2015-09-15T12:35:38.403Apologies! I omitted a key element of my question, that I was only concerned with

perspective projection. I'm very rusty in this field and its terminology after many years away from it, yet I remain interested. By the way a [tag:perspective] would be a worthwhile addition to the site for questions such as this. – hippietrail – 2015-09-15T14:55:59.8701In that case I will promote my comments to an answer... – Simon F – 2015-09-15T15:06:46.273

1you need to add a constraint. fisheye is also a perspective projection, and you won't get ellipses. the constraint you need is linearity. – v.oddou – 2015-09-16T01:16:17.283

@v.oddou: Thanks for your help with the terminology. Would that result in something like "projected into linear perspective" or something else? – hippietrail – 2015-09-16T05:18:15.043

1I would rather say something like "where the projection is a linear application". There might be some shortcut term for this, like "linear epimorphism" or something, but I long forgot that. – v.oddou – 2015-09-16T06:14:06.193

@v.oddou: I've tweaked the wording of the question based on your advice. – hippietrail – 2015-09-16T06:21:29.973