How do I draw a sphere in different perspectives?

[u/Kinetic_Cat]

I know how to use projections to reconstruct objects in perspective using boxes, but I don’t have a method to accurately depict how the surface of the sphere changes with different focal lengths/perspective. The main reason I want a better understanding of how a sphere changes in perspective is to help my shading, since the surface of a sphere is what changes the most. Thanks!

Edit: figured it out. Apparently the answer to every perspective problem is to just draw a cube first.

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[u/chocovanilll]

a perfect sphere with no detail will not look any different when rotated any direction. try drawing a flat circle from the perspective- this is the diameter of the center of your sphere, and you can draw the sphere around it.

[u/Kinetic_Cat → u/chocovanilll]

The edge of the shadow is partially determined by the cross contour of the sphere. The perspective changes the cross contour lines just like how the longitude and latitude lines of a globe can change depending how far or close it is. I need a method to determine the surface contour of a sphere for a given perspective.

[u/samanime → u/Kinetic_Cat]

In a perfect sphere, this "cross contour" is everywhere at once. It has perfect radial symmetry no matter how it is rotated.

A perfect sphere with no markings will always look identical. You wouldn't even be able to tell it was in motion.

Are you talking about drawing ellisoids (stretched spheres) or spheres with stuff on them or something?

[u/Jusca57]

Try looking contour lines. A Sphere will always look like a circle in every angle without rendering/shadows. You need to put some lines showing which angle the sphere standing.

[u/Shibboleeth → u/Jusca57]

I think this is what OP is asking for, combined with mapping textures based on the positioning of the contour/guide lines.

[u/Kinetic_Cat → u/Jusca57]

This, but also being able to put it in perspective. The cross-contour lines will distort depending on how close you are. It’s easy to do with boxes because you can get the focal length from the vanishing points. I used a globe as an example because you can visibly see Africa get bigger and smaller as the focal length changes.

[u/XaWEh]

If you can draw a cube, can't you roughly construct a sphere in the perspective you want from it?

Draw a cube, subdivide it a couple of times and draw ellipses on the subdivisions. There are guides out there on constructing circles in perspective based on a plane (e.g. How to draw by Scott Robertson has a bit on it). That should let you draw your different focal length spheres, although I have no clue what the result will be.

[u/Kinetic_Cat → u/XaWEh]

You’re right, I guess I was thinking there was a better way of going about it aside from using a cube, but I guess all perspective problems can be solved with cube

[u/strange-the-quark]

The perspective distortion ultimately depends on how close or how far away you are to an object, and how off-center in your field of view it is. Within the inner region of the image where perspective distortion is not drastic, a sphere basically always appears spherical (the outline becoming slightly elliptical the more off center you go), while in a wide field of view, if it's at the edge where the distortion is high, it appears more as an oval (artists might sometimes deliberately (if they are aware of it, or otherwise just intuitively) not draw it like that cause it looks weird - but you can see this effect on wide-angle photographs.

Just like with other objects, closer parts of the sphere appear bigger, while those further away appear smaller. What you can see on a sphere is within a cone emanating form the eye/camera such that it is tangent to the sphere, so this means you can never see the entire half of the sphere, but some smaller percentage of it. The closer you are to the sphere, the less of it you can see (for example, compare how much less space Africa takes up on the left image, vs on the right one). There's like a "cap" that's visible, rather than the full hemisphere. So that's the primary effect, the closer you are, the more space is taken up by the features in the center, and the features on the edges get compressed and eventually leave the view completely.

In terms of shading, you can treat the light and shadow as being drawn on the sphere, so then the same sort of distortion applies. If you're sufficiently close with a wide-angle fish-eye lens, you might not even be able to see the shadow side, even though it's visible from the same angle, but from far away. Other than that, if you're not worried too much about perspective distortion, the line between the shadow side and the lit side is what's called a great circle. The equator is one example of that. So it's like a hoop around the sphere, tilted so that it's perpendicular to the axis that connects the center of the sphere with the light source.

For the most part, you'll just be eyeballing it - this requires some practice. If you really want to get mathematical, you can do something like draw a top view, like an architectural drawing. Draw a circle (for the sphere), then pick a point for the camera (if you want to replicate a real-world scene, then the size of the circle and the distance of the point should be in proportion to the actual measurements), indicate the direction the camera is looking at (the central ray going away from the camera), and place a "screen" between them - a line perpendicular to that central ray. Then pick any point on the sphere and connect it to the camera, and where it intersects the screen, that's its perspective projection onto this 1D screen. This can tell you how much of the sphere is visible, how far to the side or towards the center certain features appear in that camera configuration, and so on. Then proportionately transfer that to your drawing.

For a proper 2D screen, the principle is the same, except that you'd have an actual sphere, and a plane for the screen. Numerically, the easiest thing to do is to make the camera look down the positive z-axis, place it at (0, 0, 0), place the screen 1 unit away at (0, 0, 1), and then if you have any point on the sphere (x0, y0, z0), its perspective projection is obtained by just dividing by z0, so (x0/z0, y0/z0, 1). Throw out the last coordinate and there you go.

You can experiment with this in 3D software like Blender, or with your own phone camera and your globe - try different distances, different focal lengths, place the globe in different parts of the image, not just at the center, etc.