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SelfCAD Explores 3D Transformations in Computer Graphics

New York, United States, December 14, 2023 – Can you see an object in 3D without 3D glasses? If you think you cannot, read on to understand what 3D transformations are.

What are 3D transformations in Computer Graphics?

In the broadest sense, a 3D model is a mathematical representation of a real-world object that takes up space. In more concrete words, a 3D model is composed of a description of its shape and how it seems to be colored. Three-dimensional (3-D) transformation changes how a three-dimensional (3-D) item appears concerning its original position by changing its physical characteristics using various transformational techniques, such as translation, scaling, rotation, shear, etc.

Computer graphics make it possible to observe objects from many perspectives. An architect can look at a building from various perspectives. Charts and topographical maps can have their sizes altered by a cartographer. The numbers can, therefore, be kept in memory if graphic images are encoded as numbers.

Mathematical operations referred to as transformations are used to change these numbers. Using a computer to draw is intended to allow the user to observe an object from several perspectives while also allowing them to transform an object’s size or shape. Each transformation stands alone. It might have a special name or symbol to identify it. It is possible to join two transformations to create a single transformation.

In geometric transformations, the actual object is altered in relation to the background or coordinate system. Geometric modifications made to each point of the object define the mathematical description of this viewpoint. The object is kept stable throughout a coordinate transformation while the coordinate system is altered in relation to the object. Coordinate transformations are used to achieve this result.

While carrying out a 3D transformation, the following properties of the model have to be maintained: The lines of the model have to be preserved, the parallelism of the model must be preserved, and Proportional Distances must be preserved.

What are the different kinds of 3D representations?

Considering whether the surface or the volume of an object is represented can help classify a 3D object’s representation.

  • Boundary-based representation: The 3D object’s surface is depicted. This illustration is also known as a “b-rep.” Common representations of this type include polygon meshes, implicit surfaces, and parametric surfaces, which you will go through in the following sections.
  • Volume-based representations: The 3D object’s volume is represented. Voxels and constructive solid geometry (CSG) are frequently utilized to represent volumetric data.

What is a transformation Matrix?

A fundamental tool for transformation is the transformation matrix. The coordinates of the items are multiplied by a matrix with n x m dimensions. Transformation typically involves the use of 3 x 3 or 4 x 4 matrices. Use the following matrix for various operations as an example.

An equation system can be represented as a matrix by isolating the equation’s coefficients from its variables. We can convert the general rotational equation into a matrix. You have to be aware that the fn values are the “coefficients” of the equation terms, and the x, y, and z values are our “variables.” By selecting the right fn values, we will generate the transformation and then apply it to each vertex in the model to change the graphics model for a computer graphics scene. For rendering a single animation frame, the fn values are fixed, but they often vary for the following frame. A set of equations representing the model is given below:

In the matrix form, it will look like this:

What are the different types of 3D transformations?

●     Translation

It alters a 3-D object’s coordinates to get it closer to its original place. With a given 3D transformation matrix, as shown in the graphic below, where Dx, Dy, and Dz are the translation distances, let P(x, y, z) be a point in 3D space across which we want to execute the translation transformation operation. After applying the translation operation, the point’s new location would be as follows:, – \, \hspace{4.5cm} \textbf{P'[x’, y’, z’, 1] = P[x, y, z, 1].T[x, y, z]}

X, Y, and Z axes are present in a 3D translation process. Any object can be moved from one location to another without affecting its shape. Squares, Polygons, Rectangles, and Lines can all be translated into three dimensions. A before and after image of the translation process is as follows

  • Scaling Transformation

Resizing a 3D object is done by applying Sx, Sy, or Sz scaling factors to change the object’s dimension in any of the x, y, or z directions. When applying scaling transformations to a fixed point, the sequences listed below take place: Translation of the fixed point to the origin. The thing has been scaled. The fixed point is moved back to where it started. As seen below, you either increase or decrease the object’s dimensions throughout the scaling procedure. To scale an object, multiply its original coordinates by the scaling factor until the desired result is obtained.

●     3D Rotation

Compared to the rotation transformation in 2D, rotation in 3D is more complex since it involves dealing with three axes (x, y, z). Compared to the rotation transformation in 2D, rotation in 3D is more complex since it involves dealing with three axes (x, y, z). Rotation happens about the three axes, as follows:

  • About the X-axis: This type of rotation keeps the x coordinate constant while rotating the object parallel to the x-axis (principal axis), changing only the other two coordinates, y and z. Imagine rotating a point in 3D space with initial coordinates P(x,y,z) parallel to the main axis (x-axis).
  • About the Y-axis: The object is rotated in this way so that it is parallel to the y-axis, the major axis, with the y coordinate remaining constant and the other two coordinates, x and z, only changing.
  • About the Z-axis: The object is rotated parallel to the z-axis (primary axis), with the z coordinate remaining constant and the other two coordinates, x and y, only changing.

●     Shearing Transformation

The shearing transformation is the same as what we observe in 2D space, but in 3D, we must also include the x, y, and z axes, whereas, in 2D, we just consider the x and y axes. Slanting an object in 3D space in the x, y, or z direction is shearing. The shape of the thing is altered via shearing. As we are talking about 3D space, shearing can also be done in any of the following three directions. The shear transformation is the name for a transformation that tilts an object’s shape. Similar to 2D shear, we can shear an object in 3D along the X, Y, or Z axes. The different shearing transformations are listed below.

  • Shearing in X direction: Y and Z’s coordinates are altered while X’s coordinate stays the same.
  • Shearing in Y direction: While X and Z’s coordinates are altered, Y’s coordinate stays the same.
  • Shearing in Z direction: Here, the coordinates of X and Y are altered while Z’s coordinate remains the same.

●     Reflection transformation

Reflection in 3D space is quite similar to reflection in 2D space, with the exception that we must deal with three axes in 3D. (x, y, z). Reflection is nothing more than an object’s mirror reflection. In 3D space, reflections of three different types are possible:

  • Reflection on the X-Y plane:
  • Reflection on the Y-Z plane.
  • Reflection on the X-Z plane.

Stunning visual effects and immersive experiences with 3D transformation

Through the use of numerous transformational techniques, such as translation, scaling, rotation, shear, etc., three-dimensional (3-D) transformation modifies how a three-dimensional (3-D) item appears in relation to its initial position. With computer graphics, it is possible to view an item from a variety of angles. So, if graphic images are encoded as numbers, the numbers may be stored in memory. These numbers are altered via transformations, which are mathematical procedures.

The purpose of using a computer to sketch is to allow the user to change the size or shape of an object as well as see it from various angles. Every metamorphosis is independent. It might be distinguished by a unique name or mark. Two transformations can be combined to generate a single transformation. The real object is changed in reference to the background or coordinates system in geometric transformations.

The mathematical description of this viewpoint is defined by the geometric adjustments applied to each point of the object. During a coordinate transformation, the object is kept steady while the coordinate system is changed in reference to the object. In order to get this outcome, coordinate transformations are used. The model’s lines, parallelism, and proportional distances must all be maintained during a 3D transformation.

You can see all these in action while you manipulate 3D objects in a CAD tool. You would use all these to change different aspects of the models you make so that the end result looks like the object in your imagination. If you have not yet used CAD or you have used and you would like the one that is easy to use and powerful at the same time to use to work on your projects you can try SelfCAD. SelfCAD is a user-friendly 3D modeling software that has been designed for both beginners and professionals and you don’t need to have previous experience in CAD to use it.

 

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