Exploring the Mystery of 3D Coordinates: A= (X, Y, Z, 1)

In summary, the reason for having 4 parameters in projective coordinates instead of 3 is due to the use of projective geometry, where translations and rotations can be represented as matrix multiplications. This allows for simpler calculations and transformations in computer graphics.
  • #1
marensi
1
0
hello EveryBody,

In the 3D Coordinates I always find 4 parameters instead of 3.

A = (X, Y, Z, 1)

I wonder why?

thank you.
 
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  • #2
Please provide more information. We can't answer your question like this.
 
  • #3
You don't. What you have there is NOT "3 D coordinates" but, rather, "3 D projective coordinates" (the name coming from "projective geometry" that I won't go into). In "projective coordinates" the point (x, y, z) is represented by (x, y, z, 1) with the understanding that if any calculation changes that last coordinate to something other than 1, say, a (and a cannot be 0), then we interpret (x, y, z, a) as meaning (x/a, y/a, z/a, 1).

Projective coordinates are often used in computer graphics because they have the property that translations, as well as rotations, can be be written as matrix multiplications.

In ordinary 3D coordinates, a rotation, by angle [itex]\theta[/itex] around the x-axis, is given by
[tex]\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 \\ sin(\theta) & cos(\theta) & 0 \\ 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z\end{bmatrix}[/tex]
with obvious changes for rotations around the y and z axes and all rotations given by products of such matrices.

In "projective coordinates" such a rotation would be just
[tex]\begin{bmatrix}cos(\theta) & -sin(\theta) & 0 & 0 \\ sin(\theta) & cos(\theta) & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ 1\end{bmatrix}[/tex]

The translation, that moves (x, y, z, 1) to (x+ a,y+ b, z+ c, 1), in projective coordinates, is given by
[tex]\begin{bmatrix}1 & 0 & 0 & a \\ 0 & 1 & 0 & b \\ 0 & 0 & 1 & c \\ 0 & 0 & 0 & 1\end{bmatrix}\begin{bmatrix}x \\ y \\ z \\ 1\end{bmatrix}[/tex]
 
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1. What are 3D coordinates and how do they work?

3D coordinates refer to a system of locating points in 3-dimensional space using x, y, and z coordinates. They work by assigning values to each of these axes, with the x-axis representing horizontal movement, the y-axis representing vertical movement, and the z-axis representing depth or distance from the origin point.

2. How are 3D coordinates used in scientific research?

3D coordinates are commonly used in scientific research, particularly in fields such as astronomy, physics, and engineering. They are used to accurately measure and map out the position, movement, and relationships of objects in 3-dimensional space, allowing for more precise analysis and predictions.

3. What is the significance of the "1" in the 3D coordinate system?

The "1" in the 3D coordinate system represents the homogeneous coordinate, which is used to represent points in a projective space. This allows for easier mathematical operations and transformations on the points, making it a useful tool in 3D graphics and computer graphics.

4. Can 3D coordinates be used to represent objects in the real world?

Yes, 3D coordinates can be used to represent objects in the real world. By assigning appropriate values to the x, y, and z axes, the coordinates can accurately describe the position and orientation of physical objects in 3-dimensional space.

5. What are some challenges associated with using 3D coordinates?

One challenge with using 3D coordinates is the potential for human error in measurement and recording. Additionally, converting between different coordinate systems, such as Cartesian and polar coordinates, can be complex and require advanced mathematical knowledge. Another challenge is accurately representing curved surfaces or objects with irregular shapes using a system that is based on straight axes.

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