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OpenGL Camera

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Related Topics: OpenGL Transform, OpenGL Projection Matrix, Quaternion
Download: OrbitCamera.zip, trackball.zip

Overview

OpenGL Eye Space
OpenGL camera is always at origin and facing to -Z in eye space

OpenGL doesn't explicitly define neither camera object nor a specific matrix for camera transformation. Instead, OpenGL transforms the entire scene inversely to a space, where a fixed camera is at the origin (0,0,0) and always looking along -Z axis. This space is called eye space.

Because of this, OpenGL uses a single GL_MODELVIEW matrix for both object transformation to world space and camera (view) transformation to eye space.

You may break it down into 2 logical sub matrices;
OpenGL ModelView matrix combined

That is, all objects in a scene are transformed with Mmodel first, then the entire scene is transformed reversely with Mview. In this page, we will discuss only Mview for camera transformation in OpenGL.

LookAt

gluLookAt() is used to construct a viewing matrix where a camera is located at the eye position eye position and looking at (or rotating to) the target point target position. The eye position and target are defined in world space. This section describes how to implement the viewing matrix equivalent to gluLookAt().

Camera's looAt transformation consists of 2 transformations; translating the whole scene inversely from the eye position to the origin, and then rotating the scene with reverse orientation, so the camera is positioned at the origin and facing to the -Z axis.
view matrix of lookAt

Camera lookAt animation
OpenGL camera's lookAt() transformation

Suppose a camera is located at (2, 0, 3) and looking at (0, 0, 0) in world space. In order to construct the viewing matrix for this case, we need to translate the world to (-2, 0, -3) and rotate it -33.7 degree along Y-axis. As a result, the virtual camera becomes facing to -Z axis at the origin.

The translation part of lookAt is easy. You simply move the camera position to the origin. The translation matrix MT would be (replacing the 4th coloumn with the negated eye position);
lookAt's translation matrix

left, up and forward vector
Left, Up and Forward vectors from target to eye

The rotation part of lookAt is much harder than translation because you have to calculate 1st, 2nd and 3rd columns of the rotation matrix all together.

First, compute the normalized forward vector forward vector from the target position vt to the eye position ve of the rotation matrix. Note that the forward vector is from the target to the eye position, not eye to target because the scene is actually rotated, not the camera is.
forward vector

Second, compute the normalized left vector left vector by performing cross product with a given camera's up vector. If the up vector is not provided, you may use (0, 1, 0) by assuming the camera is straight up to +Y axis.
left vector

Finally, re-calculate the normalized up vector up vector by doing cross product the forward and left vectors, so all 3 vectors are orthonormal. Note we do not normalize the up vector because the cross product of 2 perpendicular unit vectors also produces a unit vector.
up vector

These 3 basis vectors, left vector, up vector and forward vector are used to construct the rotation matrix MR of lookAt, however, the rotation matrix must be inversed. Suppose a camera is located above a scene. The whole secene must rotate downward inversely, so the camera is facing to -Z axis. In a similar way, if the camera is located to the left of the scene, the scene should rotate to right in order to align the camera to -Z axis. The following diagrams show why the rotation matrix must be inversed.
Camera pitch animation
The scene rotates downward if the camera is above
Camera yaw animation
The scene rotates to right if the camera is at left

Therefore, the rotation part MR of lookAt is finding the rotation matrix from the target to the eye position, then inverse it. And, the inverse matrix is equal to its transpose matrix because it is orthogonal which each column has unit length and perpendicular to the other column.
Ratation matrix of lookAt

Finally, the view matrix for camera's lookAt transform is multiplying MT and MR together;
OpenGL LookAt matrix

Here is C++ snippet to construct the view matrix for camera's lookAt transformation. Please see the details in OrbitCamera.cpp.


// dependency: Vector3 and Matrix4
struct Vector3
{
    float x;
    float y;
    float z;
};

class Matrix4
{
    float m[16];
}

///////////////////////////////////////////////////////////////////////////////
// equivalent to gluLookAt()
// It returns 4x4 matrix
///////////////////////////////////////////////////////////////////////////////
Matrix4 lookAt(Vector3& eye, Vector3& target, Vector3& upDir)
{
    // compute the forward vector from target to eye
    Vector3 forward = eye - target;
    forward.normalize()                  // make unit length

    // compute the left vector
    Vector3 left = upDir.cross(forward); // cross product
    left.normalize();

    // recompute the orthonormal up vector
    Vector3 up = forward.cross(left);    // cross product

    // init 4x4 matrix
    Matrix4 matrix;
    matrix.identity();

    // set rotation part, inverse rotation matrix: M^-1 = M^T for Euclidean transform
    matrix[0] = left.x;
    matrix[4] = left.y;
    matrix[8] = left.z;
    matrix[1] = up.x;
    matrix[5] = up.y;
    matrix[9] = up.z;
    matrix[2] = forward.x;
    matrix[6] = forward.y;
    matrix[10]= forward.z;

    // set translation part
    matrix[12]= -left.x * eye.x - left.y * eye.y - left.z * eye.z;
    matrix[13]= -up.x * eye.x - up.y * eye.y - up.z * eye.z;
    matrix[14]= -forward.x * eye.x - forward.y * eye.y - forward.z * eye.z;

    return matrix;
}

Rotation

Shifting

Forwarding

Example: OpenGL Orbit Camera

OpenGL OrbitCamera

Download: OrbitCamera.zip

Example: Trackball

OpenGL Trackball

Download: trackball.zip

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