Two years ago I read a blog entry by Pete Shirley about left-handed vs. right-handed coordinates. I started to have a go at explaining these as simply as I could, but kept putting it off, to avoid saying something stupid or confusing. Having just dealt with this issue yet again at work, it’s time to write down my mental model.

One problem in thinking about this area is that there are two places where we care about them: world coordinates (where stuff is in space) and view coordinates (what we use for the view and perspective matrices). So, this first post will be just about world coordinates, as a start. Basic, but let’s get it down to begin.

The way I think about RH vs. LH for world space is there’s an objective reality out there. You are trying to define where stuff is in that reality. You stand on a plane and decide that looking East is the X+ axis, looking North is the Y+ axis (typical Cartesian coordinates). For the Z+ axis you decide that altitudes are positive numbers. That’s an RH coordinate system, and that’s why it’s the one used by most modeling packages, AFAIK (please do let me know if there are any LH modelers). We all likely know about how the right hand is used to explain the counterclockwise twist the three axes form, the right-hand rule. I was also happy to see on this same page how to label your two fingers and thumb to show the coordinate system.

You meet with Marvin the Moleman. He likes Y+ North, X+ East, just like you, but Z+ for him is downwards, his numbers increase as he digs his holes. He’s LH. So he hands you a model of his mole-lair, fully modeled in 3D. Fine, the transform to the RH space you like is a Z axis reflection, i.e., negate the Z coordinate and, as needed, normal values. He also gives you a 2D textured rectangle showing the floor plan, a 2D object. Viewing his dataset from above, your and his XYZ coordinates (and UV coordinates) happen to exactly match, the Z flip does nothing to these coordinates since Z is 0. You flip only the rectangle’s normal direction.

There are an infinite number of ways to transform between LH and RH, not just negate Z. A plane with any orientation and location can be used to mirror the vertices of the model; some planes are just more useful and convenient than others. A quick rule is that negating one axis or swapping two axes transforms from one coordinate system to the other.

One interesting property of such conversions is that, even though the normals get flipped along some plane (or perhaps I should say, because the normals get flipped), clockwise order of the vertices is not affected by conversion between these two model coordinate systems. Which is counterintuitive, on the face of it: if, for example, you do a planar mirror transform of a model so that you can render it again as an object reflected in a mirror (p. 386 on in RTR3), the mirroring matrix most definitely does change the ordering of the vertices. A clock seen in a mirror is reversed in the direction its hands travel.

The point is simply this: LH and RH are indeed just two ways of describing the same underlying world space. Conversion between the two does not change the world. A clock in the real world will always have its hands move in a clockwise direction, regardless of how you describe that world. I find this “conversion that does not modify clockwise” operation to be like the Zen koan, “It is not the wind that moves. It is not the flag that moves. It is your mind that moves.”

One last bit I found interesting: latitude/longitude. Typically we describe a location on the earth as lat/long/altitude, with North positive for latitude, East positive for longitude. So lat/long/altitude is left-handed, assigning them in this XYZ order. But, I’ve also seen such coordinates listed in longitude/latitude order, e.g., TerraServer USA uses this order. Which is right-handed, since the X and Y coordinates are swapped. In this case all the values are the same, but it’s simply the ordering that changes the handedness.

My next posting on this subject will be about LH vs. RH for viewing vs. world coordinates, which is where the real confusion comes in.