[Reprint] Isometric (oblique 45 degrees) game and math

Original link: http://www.javaeye.com/articles/1225

1984 European players were shocked by a then-released tour Knight lore. This 2d game incredibly realistic simulation of the 3d environment, and then the game's release platform is ZX Spectrum, 48K of memory. This technique is isometric perspective (conformal projection). Later, the Big Pineapple, Final Fantasy and other games have adopted this projection technology.
Conformal projection is a kind of projection that remains unchanged before and after the angle projection of two differential segments at any point on the projection plane within a certain range. A type of projection that remains unchanged before and after an angle projection consisting of two differential segments at any point. is the angle and shape to keep the correct projection, also known as a positive projection. This type of projection differs from the normal perspective projection of true 3D, and does not scale to the proximity of the object to the camera, and the X, y, and Z axes are projected at a 120-degree angle. But for the computer to render pixel image graphics features, and development convenience, many games are actually using a two-conformal projection.
You can refer to this formula by changing the isometric coordinates to screen coordinates:

Transform from screen coordinates to isometric coordinates:

The following articles are excerpted from http://www.compuphase.com/axometr.htm:
The figure below shows a cube in the isometric projection as defined by ISO 5456-3.the first object from the left in the F Igure is the cube unadorned; The second object is the same cube with angles and measures annotated around it. The third and fourth graphics is the top and side views of the perspective scene and they give the camera position that F Its the Perspective view. The camera position is a would feed into a 3D renderer (or Ray Tracer) to create the sprites or tiles for the Isome Tric projection.

Computer games with isometric maps is often tile-based. To make tiles match, the game designer must take into account how diagonal lines is plotted in discrete steps (Bresenham and all that). As it turns out, a line at $ degrees (sine is 0.5) produces steps this is too irregular. A angle where the tangent is 0.5 does has a nice regular pattern:two steps to the right and one step up. Thus, the isometric projection used by most games tilt the X-and z-axes with approximately degrees (the exact angle is "Arctangent (0.5)"). By the, because the tangent of the angle of the rhombus are 0.5, the rhombus is twice as wide as it's high. Many sources mention a 1:2 scale for isometric perspectives. (To make the edges of the rhombuses match, the width of the rhombus should is a multiple of four pixels and the height sho Uld is one pixel less than½width. In other words, the height:width ratio of a rhombus are usually not exactly 1:2, but rather near 1:2.1. This makesNo difference for the principles of the isometric perspective.)
Again, the figure of below shows what's this isometric projection looks like.

Flash game with conformal projection of the Playfish restaurant (requires FQ):

[Reprint] Isometric (oblique 45 degrees) game and math