3-D Transformation in Computer Graphics
Three-dimensional (3-D) transformations play a crucial role in computer graphics, allowing us to manipulate objects in a three-dimensional space. These transformations include translation, rotation, scaling, and shearing, each altering the position, orientation, and size of objects in the 3-D world.
Translation
Translation involves moving an object from one location to another in 3-D space. It shifts every point of the object by a specified distance along the x, y, and z axes. For example, to translate an object 5 units to the right, 3 units upwards, and 2 units towards the viewer, we modify the coordinates of each vertex accordingly.
newX = oldX + dx;
newY = oldY + dy;
newZ = oldZ + dz;
Rotation
Rotation changes the orientation of an object around a specified axis. It can be performed about the x-axis, y-axis, or z-axis, altering the object's appearance in the 3-D space. For instance, to rotate an object 45 degrees clockwise around the y-axis, we apply trigonometric functions to calculate the new coordinates of each vertex.
newX = oldX * cos(angle) - oldZ * sin(angle);
newY = oldY;
newZ = oldX * sin(angle) + oldZ * cos(angle);
Scaling
Scaling modifies the size of an object along the x, y, and z axes. It can either enlarge or shrink the object based on scaling factors. For example, to double the size of an object, we multiply the coordinates of each vertex by 2. Scaling is useful for creating perspective effects and adjusting the dimensions of objects.
newX = oldX * scaleX;
newY = oldY * scaleY;
newZ = oldZ * scaleZ;
Shearing
Shearing distorts the shape of an object by skewing it along one of the axes while leaving the other axes unchanged. It is often used in computer graphics to simulate effects such as perspective distortion or creating slanted objects. Shearing involves modifying the coordinates of vertices based on the desired shearing factors.
newX = oldX + shearX * oldY;
newY = oldY + shearY * oldX;
newZ = oldZ;
In conclusion, 3-D transformations are essential techniques in computer graphics for manipulating objects in three-dimensional space. Whether it's translating, rotating, scaling, or shearing, these transformations enable us to create dynamic and visually appealing graphics in various applications.