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Overview

Geometric data such as vertex positions and normal vectors are transformed via Vertex Operation and Primitive Assembly operation in OpenGL pipeline before raterization process.

OpenGL vertex transformation

Object Coordinates

It is the local coordinate system of objects and is initial position and orientation of objects before any transform is applied. In order to transform objects, use glRotatef(), glTranslatef(), glScalef().

Eye Coordinates

It is yielded by multiplying GL_MODELVIEW matrix and object coordinates. Objects are transformed from object space to eye space using GL_MODELVIEW matrix in OpenGL. GL_MODELVIEW matrix is a combination of Model and View matrices (). Model transform is to convert from object space to world space. And, View transform is to convert from world space to eye space.

Note that there is no separate camera (view) matrix in OpenGL. Therefore, in order to simulate transforming the camera or view, the scene (3D objects and lights) must be transformed with the inverse of the view transformation. In other words, OpenGL defines that the camera is always located at (0, 0, 0) and facing to -Z axis in the eye space coordinates, and cannot be transformed. See more details of GL_MODELVIEW matrix in ModelView Matrix.

Normal vectors are also transformed from object coordinates to eye coordinates for lighting calculation. Note that normals are transformed in different way as vertices do. It is mutiplying the tranpose of the inverse of GL_MODELVIEW matrix by a normal vector. See more details in Normal Vector Transformation.

Clip Coordinates

The eye coordinates are now multiplied with GL_PROJECTION matrix, and become the clip coordinates. This GL_PROJECTION matrix defines the viewing volume (frustum); how the vertex data are projected onto the screen (perspective or orthogonal). The reason it is called clip coordinates is that the transformed vertex (x, y, z) is clipped by comparing with ±w.

See more details of GL_PROJECTION matrix in Projection Matrix.

Normalized Device Coordinates (NDC)

It is yielded by dividing the clip coordinates by w. It is called perspective division. It is more like window (screen) coordinates, but has not been translated and scaled to screen pixels yet. The range of values is now normalized from -1 to 1 in all 3 axes.

Window Coordinates (Screen Coordinates)

It is yielded by applying normalized device coordinates (NDC) to viewport transformation. The NDC are scaled and translated in order to fit into the rendering screen. The window coordinates finally are passed to the raterization process of OpenGL pipeline to become a fragment. glViewport() command is used to define the rectangle of the rendering area where the final image is mapped. And, glDepthRange() is used to determine the z value of the window coordinates. The window coordinates are computed with the given parameters of the above 2 functions;

glViewport(x, y, w, h);

glDepthRange(n, f);

The viewport transform formula is simply acquired by the linear relationship between NDC and the window coordinates;

OpenGL Transformation Matrix

OpenGL Transform Matrix

OpenGL uses 4 x 4 matrix for transformations. Notice that 16 elements in the matrix are stored as 1D array in column-major order. You need to transpose this matrix if you want to convert it to the standard convention, row-major format.

OpenGL has 4 different types of matrices; GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE, and GL_COLOR.

Geometric data such as vertex positions and normal vectors are transformed via Vertex Operation and Primitive Assembly operation in OpenGL pipeline before raterization process.

OpenGL vertex transformation

Object Coordinates

It is the local coordinate system of objects and is initial position and orientation of objects before any transform is applied. In order to transform objects, use glRotatef(), glTranslatef(), glScalef().

Eye Coordinates

It is yielded by multiplying GL_MODELVIEW matrix and object coordinates. Objects are transformed from object space to eye space using GL_MODELVIEW matrix in OpenGL. GL_MODELVIEW matrix is a combination of Model and View matrices (). Model transform is to convert from object space to world space. And, View transform is to convert from world space to eye space.

Note that there is no separate camera (view) matrix in OpenGL. Therefore, in order to simulate transforming the camera or view, the scene (3D objects and lights) must be transformed with the inverse of the view transformation. In other words, OpenGL defines that the camera is always located at (0, 0, 0) and facing to -Z axis in the eye space coordinates, and cannot be transformed. See more details of GL_MODELVIEW matrix in ModelView Matrix.

Normal vectors are also transformed from object coordinates to eye coordinates for lighting calculation. Note that normals are transformed in different way as vertices do. It is mutiplying the tranpose of the inverse of GL_MODELVIEW matrix by a normal vector. See more details in Normal Vector Transformation.

Clip Coordinates

The eye coordinates are now multiplied with GL_PROJECTION matrix, and become the clip coordinates. This GL_PROJECTION matrix defines the viewing volume (frustum); how the vertex data are projected onto the screen (perspective or orthogonal). The reason it is called clip coordinates is that the transformed vertex (x, y, z) is clipped by comparing with ±w.

See more details of GL_PROJECTION matrix in Projection Matrix.

Normalized Device Coordinates (NDC)

It is yielded by dividing the clip coordinates by w. It is called perspective division. It is more like window (screen) coordinates, but has not been translated and scaled to screen pixels yet. The range of values is now normalized from -1 to 1 in all 3 axes.

Window Coordinates (Screen Coordinates)

It is yielded by applying normalized device coordinates (NDC) to viewport transformation. The NDC are scaled and translated in order to fit into the rendering screen. The window coordinates finally are passed to the raterization process of OpenGL pipeline to become a fragment. glViewport() command is used to define the rectangle of the rendering area where the final image is mapped. And, glDepthRange() is used to determine the z value of the window coordinates. The window coordinates are computed with the given parameters of the above 2 functions;

glViewport(x, y, w, h);

glDepthRange(n, f);

The viewport transform formula is simply acquired by the linear relationship between NDC and the window coordinates;

OpenGL Transformation Matrix

OpenGL Transform Matrix

OpenGL uses 4 x 4 matrix for transformations. Notice that 16 elements in the matrix are stored as 1D array in column-major order. You need to transpose this matrix if you want to convert it to the standard convention, row-major format.

OpenGL has 4 different types of matrices; GL_MODELVIEW, GL_PROJECTION, GL_TEXTURE, and GL_COLOR.

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