Windows Phone 7 Game Development : The World of 3D Graphics - Perspective Projection

In addition to this obvious size effect, the more subtle effects of perspective are picked up intuitively by the brain and add a substantial feeling of depth to the rendered scene. The sides of a cube will very slightly narrow due to the effects of perspective as they increase in distance from the viewer, allowing the brain to automatically determine the exact position in which the cube is situated.

1. The Viewing Frustum

When we use a perspective projection in XNA, we create as part of the Effect object initialization a three-dimensional volume known as a viewing frustum. The shape of the frustum is that of a rectangular cone with its tip cut off. A demonstration of such a frustum is shown in Figure 1.

Figure 1. A diagram showing a 3D viewing frustum

The frustum can be visualized in the real world by imagining yourself looking through a window. Outside the window you can see the ground and various objects. The farther away into the distance you look, the wider the area that you can see. Objects that are too far to the side, above, or below will be hidden by the window frame.

Objects that fall inside the volume described by the frustum are visible to the camera (and would be visible through the window). Objects that fall outside the frustum volumes are hidden from the camera (and would not be able to be seen through the window).

The near and far clip planes are also taken into account when deciding whether objects are visible. Objects nearer to the camera than the near clip plane are deemed to be too close and are excluded from rendering. Similarly, objects farther than the far clip plane are too far away to be seen and are once again excluded.

When XNA transforms the objects that fall inside the frustum from the 3D space in which we have defined our world into the 2D space that is actually presented on the screen, it takes into account how much of the width and height of the frustum is filled by any particular object. An object will occupy a greater proportion of the frustum when it is positioned toward the near clip plane than it will at the far clip plane (see Figures 2 and 3). Figure 2 shows two identically sized objects within the viewing frustum. Figure 3 shows the same scene after the perspective projection has taken place to transform the scene into two dimensions for display on the screen. Note that the object at the near clip plane appears substantially larger than the one at the far clip plane.

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

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

In addition to the clip planes, the frustum is defined by two more pieces of information: the viewing angle and the aspect ratio.

The viewing angle defines the angle, in degrees, between the camera and the upper edge of the frustum (the angle on the y axis). Changing this angle will make the overall shape of the frustum expand or compress, causing the apparent reduction in size of objects farther away to be increased or decreased.

Figure 4 shows two viewing frustums from the side: the first with a viewing angle of 45 degrees; the second with 22.5 degrees. The distance of the near and far clip planes is the same in both cases.

Figure 4. Two viewing frustums, one with a 45-degree viewing angle (left) and one with a 22.5-degree angle (right)

Consider how objects that fall into these two frustums will be projected. On the left, objects can deviate further from the center of the frustum and still be seen by the camera. Objects that are farther away will become rapidly smaller as their size relative to the extent of the frustum becomes less and less. On the right, objects farther from the center will leave the frustum more quickly and so will disappear off the edge of the screen. Distant objects will appear larger than with the first frustum as they occupy a greater proportion of the frustum's area.

Exactly what you should specify for the viewing angle will potentially vary from one game to the next. An angle of 45 degrees is usually a safe value. Setting the angle too low can make it appear that everything is closer to the player than it really is, which can result in the game feeling uncomfortable to play.

The second piece of information that the viewing frustum requires is the aspect ratio, which is calculated by dividing the display width by its height. The aspect ratio allows the viewing angle on the x axis to be calculated by XNA in response to the explicit angle that we provided for the y axis. The aspect ratio, together with the viewing angle and the distance of the clip planes, provides everything that is needed to fully describe the frustum.

2. Defining the Viewing Frustum in XNA

XNA actually performs the perspective transformation with the use of another matrix. Listing 1 is present to set up the default projection matrix.

Example 1. Creating the viewing frustum matrix

// Calculate the screen aspect ratio
float aspectRatio =
        (float)GraphicsDevice.Viewport.Width / GraphicsDevice.Viewport.Height;
// Create a projection matrix
Matrix projection = Matrix.CreatePerspectiveFieldOfView(MathHelper.ToRadians(45),
                                                        aspectRatio, 0.1f, 1000.0f);

					  

You should be able to spot all the information that we have discussed as being required for the viewing frustum. The work of generating the projection matrix is performed by the static Matrix.CreatePerspectiveFieldOfView function. The parameters that it expects to be passed are as follows, in the following order:

• fieldOfView: the viewing angle for the projection

• aspectRatio: the aspect ratio of the display width/height

• nearPlaneDistance: the near clipping plane distance

• farPlaneDistance: the far clipping plane distance

In Listing 1, a viewing angle of 45 degrees is specified, along with the aspect ratio calculated from the game's ViewPort, and near and far clipping planes of 0.1 and 1000, respectively.

When rendering, XNA first calculates the positions of all the object vertices in 3D space and then uses the projection matrix to transform them into 2D coordinates to display on the screen.

If you want to change the viewing angle (or any of the other properties of the frustum), you can simply set a new projection matrix into the effect's Projection property before rendering your objects.

The Perspective example project in this article's accompanying download shows how objects move when a perspective projection is applied. The code is very simple; it creates a number of objects with random x, y, and z coordinates. Each time the objects update, they add to their PositionZ value, moving them closer to the screen. When they get to a value of 0, they add 100 to move them back into the distance. A screenshot from the demo can be seen in Figure 5.

Figure 5. The Perspective project showing same-sized objects at different distances