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What is 3D Translation in Computer Graphics Algorithm?

Published in 3D Computer Graphics Translation 4 mins read

In computer graphics, 3D translation is fundamentally a process of moving an object from one position to another in a three-dimensional plane. It's one of the most basic and frequently used transformations applied to objects or points within a 3D scene.

Understanding 3D Translation

At its core, 3D translation shifts every point of an object by a specific displacement vector in the x, y, and z directions. Imagine you have an object, like a cube or a character, and you want to move it without rotating it or changing its size. Translation is the operation you would use.

As the reference states, 3D Translation is a process of moving an object from one position to another in a three dimensional plane. To understand this algorithm, consider a point object O that has to be moved from one position to another in a 3D plane. The algorithm determines the new coordinates of this object based on how far it needs to move along each axis.

How the Algorithm Works

The translation algorithm works by adding a translation vector to the original coordinates of a point. If a point has original coordinates (Xold, Yold, Zold), and you want to translate it by distances Tx, Ty, and Tz along the X, Y, and Z axes respectively, the new coordinates (Xnew, Ynew, Znew) are calculated using simple additions:

  • Xnew = Xold + Tx
  • Ynew = Yold + Ty
  • Znew = Zold + Tz

Here, as the reference mentions, Tx defines the distance the Xold coordinate has to be moved, and Ty defines the distance the Yold coordinate has to be moved. Tz performs the same function for the Z-axis, allowing movement into or out of the screen plane.

Mathematical Representation

In computer graphics, transformations like translation are often represented using matrices, particularly homogeneous coordinates. While translation itself is an addition, using a matrix allows it to be combined with other transformations like rotation and scaling through matrix multiplication.

For a point P(X, Y, Z), its homogeneous representation is P(X, Y, Z, 1). The translation matrix T for displacements Tx, Ty, Tz is:

| 1  0  0  Tx |
| 0  1  0  Ty |
| 0  0  1  Tz |
| 0  0  0  1  |

The new point P'(X', Y', Z') in homogeneous coordinates P'(X', Y', Z', 1) is calculated by multiplying the translation matrix by the original point's homogeneous vector:

P' = T * P

| X' |   | 1  0  0  Tx |   | X |
| Y' | = | 0  1  0  Ty | * | Y |
| Z' |   | 0  0  1  Tz |   | Z |
| 1  |   | 0  0  0  1  |   | 1 |

Performing the matrix multiplication yields:

  • X' = 1*X + 0*Y + 0*Z + Tx*1 = X + Tx
  • Y' = 0*X + 1*Y + 0*Z + Ty*1 = Y + Ty
  • Z' = 0*X + 0*Y + 1*Z + Tz*1 = Z + Tz
  • 1 = 0*X + 0*Y + 0*Z + 1*1 = 1

This confirms the simple additive calculation shown earlier, but using matrices is crucial for chaining transformations.

Translation Using Homogeneous Coordinates

Here's a simple table illustrating the transformation of a point (2, 3, 4) by a translation vector (Tx=5, Ty=-2, Tz=1) using homogeneous coordinates:

Aspect Original Point (P) Translation Matrix (T) New Point (P') Calculation New Point (P') Coordinates
Homogeneous Coordinates (2, 3, 4, 1) (See matrix above) P' = T * P (7, 1, 5, 1)
Cartesian Coordinates (2, 3, 4) Translation Vector (5, -2, 1) X' = 2+5, Y' = 3+(-2), Z' = 4+1 (7, 1, 5)

As you can see, the point originally at (2, 3, 4) is moved to (7, 1, 5) after the translation.

Practical Applications

3D translation is fundamental in countless computer graphics applications, including:

  • Positioning Objects: Placing models (characters, furniture, buildings) in a scene.
  • Animation: Moving objects or characters over time to create motion.
  • Camera Movement: Shifting the camera's position to change the view.
  • User Interaction: Dragging and dropping objects with a mouse or touch.
  • Physics Simulations: Calculating the new position of objects based on velocity.

In essence, anytime something needs to change its location in a 3D environment without changing its orientation or size, a translation algorithm is used.

Key Takeaways

  • 3D translation is the process of shifting points or objects in a 3D space.
  • It works by adding a displacement value (Tx, Ty, Tz) to the original coordinates (Xold, Yold, Zold) to get new coordinates (Xnew, Ynew, Znew).
  • Mathematically, this is often represented using matrix multiplication with homogeneous coordinates for seamless integration with other 3D transformations.
  • It's a foundational operation used widely in 3D modeling, animation, gaming, and simulation.

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