I was thinking about Zdog and my similar
<canvas> hack
the other day for Dercuano drawings, and today watching Κορη play a
slightly-3D game, it occurred to me that a little bit of
texture-mapping would go a long way to help the illusion of depth.
But both my hack and Zdog are based on, basically, spheres. How do you texture-map a sphere? The mapping from screen space to texture uv-coordinates has a couple of singularities (where the line of sight is tangent to the sphere) and so can’t be reasonably approximated with a polynomial; Babbage’s Method of Finite Differences is out of the question.
Consider the plane that includes the viewpoint and a scan line on the screen. The intersection, if it exists, between a sphere and this plane is a circle; along that circle we will find the points on the sphere that are visible in that scan line. If we consider only orthographic projection for a moment, we can use the standard midpoint algorithm for rasterizing circles to find the coordinates of the points in this circle. We may be able to use texel-sized steps if the sphere is close enough.
If we’re using a solid texture, that’s all we need — we apply the texture function (suitably strength-reduced) to the coordinates and get the colors to paint, but if we’re using a flat texture, we may need to map from 3-D space to the texture’s uv-space. Ideally we can use a polynomial fit (again, strength-reduced) to map into that space.
Adding perspective back in is a matter of stretching the middle of the scanned texture pixels while compressing the ends, some all the way to zero. We should be able to do this with a simple polynomial approximation, probably cubic, of the transformation from the orthographic view to the perspective view.
Actually, though, can’t we do this all the way through? The transformation from screen pixel space to texel space has two singularities in it, but the inverse transformation, from texel space to screen pixel space, is singularity-free and actually relatively calm. Perhaps we can just walk along the texels on the appropriate path along the sphere’s texture map, with a stride size small enough to transform at least two texels to every screen pixel, transforming each one to screen space and painting it into a screen buffer. A strength-reduced cubic spline should be perfectly adequate.