Windows Phone 7 Game Development : Orthographic Projection (part 1) - The Viewing Frustum & Defining the Orthographic Viewing Frustum in XNA

Whereas the perspective projection causes objects farther away from the camera to become smaller, an orthographic projection does not: distant objects appear at exactly the same size as near objects. This is clearly an unrealistic representation of the real world and unsuitable for any game that tries to present a lifelike approximation of a 3D environment, but it does have two specific uses.

The first of these uses is for isometric 3D games. They are generally tile-based games viewed such that the camera is rotated around and elevated from its default position. Isometric viewpoints were common in pseudo-3D games before hardware acceleration become popular. Some famous games that have used this style of 3D graphics include Q*bert, Zaxxon, Knight Lore, Marble Madness, Populous, and (more recently) Civilization III and Diablo II. Isometric games are less common these days, but do still make occasional appearances, particularly as role playing and strategy games.

The second thing that orthographic projections are useful for is creating a pixel-aligned coordinate system. Clearly with perspective projections, moving an object one unit along the x or y axis might cause it to move a different number of physical pixels along that axis, depending on how near or far the object is from the camera. Because distance makes no difference in orthographic projections, a coordinate system can be set up that exactly matches the pixels on the screen, making precise pixel-based movement much easier.

Despite their different appearance, all other aspects of 3D rendering that we have explored still hold true with orthographic projection: hidden surfaces will still be removed, objects will still be lit, and transformations and rotations will still operate exactly as with perspective projection (though rotations can look a little strange without perspective: the brain thinks they are distorting and stretching because no equivalent transformation exists in the real world).

1. The Viewing Frustum

The shape of an orthographic viewing frustum is simply a cuboid area, such as the one shown in Figure 1.

Figure 1. A diagram showing a 3D orthographic viewing frustum

Just as with a perspective projection, we still have a near and far clip plane, and objects that are rendered will still be checked to ensure that they fall within this region. Despite objects not growing or shrinking as their depth changes within a scene, the depth buffer is still used and will ensure that objects with a higher z value will appear in front objects with a lower z value.

The reason why depth has no effect on the object size is that objects at the far clip plane occupy exactly the same proportion of the plane as objects on the near clip plane.  Figure 2 shows the objects within the frustum, one at the near clip plane and one at the far clip plane just as before.

Figure 2. Two identically sized objects in the viewing frustum shown in 3D space

When these objects are transformed by the orthographic projection, they continue to appear at the same size, as can be seen in Figure 3. The proportion of the clip plane filled by the shapes is the same in both cases, and so they are not enlarged or shrunk at all.

Figure 3. The same two objects after orthographic projection into 2D

When we set up an orthographic projection, we simply tell it how many units we want it to use across the x and y axis. It will automatically stretch the rendered objects to fit within this defined set of coordinates. If the ratio of the axes does not match that of the screen, we can end up with objects becoming distorted as they are rendered. For example, if we used a range of −1 to +1 for both the x and y axes and displayed it on a 480 × 800 pixel screen, a 1-unit-square object would appear with a width of 240 pixels and a height of 400 pixels. It is clearly not square! For this reason, we generally still use the aspect ratio in our own calculations when setting the orthographic scale.

There is no need for a viewing angle to be specified for this projection because the angle is always parallel.

2. Defining the Orthographic Viewing Frustum in XNA

Just as XNA provided a useful function for creating projection matrices, so it provides another for orthographic projections—two, in fact, as you will see.

The first of these functions is the static Matrix.CreateOrthographic function. This expects the following four parameters:

• width: the number of units to display across the projection. The center point will always be 0, so providing a width of 4 will result in a frustum that extends from −2 to +2 across the x axis.

• height: the number of units to display vertically for the projection. Just as with the width, the center point will always be 0.

• zNearPlane: the near clipping plane distance.

• zFarPlane: the far clipping plane distance.

Listing 1 shows an example orthographic projection matrix being created. Its vertical size is set at 16 units, and the horizontal size is calculated from the aspect ratio to display the appropriate amount to keep the coordinate system square.

Example 1. Calculating the normals for an indexed triangle list

// Calculate the screen aspect ratio
        float aspectRatio =
                     (float)GraphicsDevice.Viewport.Width / GraphicsDevice.Viewport.Height;
        // Create a projection matrix
        Matrix projection = Matrix.CreateOrthographic(16 * aspectRatio, 16, 0, 100.0f);

					  

Let's take a look at a couple of applications of orthographic projection.