6.1 3D coordinate systems and transformations
4 min read•august 7, 2024
3D coordinate systems and transformations are crucial for spatial computing. They let us represent and manipulate objects in virtual spaces. Understanding different coordinate types and how to move things around is key to creating immersive experiences.
Mastering these concepts opens up a world of possibilities in AR and VR. From placing virtual objects in real spaces to creating complex animations, these tools are the building blocks of spatial computing applications.
Coordinate Systems
Cartesian and Homogeneous Coordinates
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- represent points in 3D space using an ordered triple (x, y, z)
- x, y, and z values determine the position along the respective axes
- Cartesian coordinates are the most common representation in computer graphics and spatial computing
- add a fourth dimension (w) to Cartesian coordinates, resulting in (x, y, z, w)
- Allows for representing points at infinity and simplifies matrix transformations
- When w = 1, the point represents a position in 3D space
- When w = 0, the point represents a direction or a point at infinity
Polar and Spherical Coordinates
- Polar coordinates represent points in a 2D plane using a distance (r) and an angle (θ)
- r is the distance from the origin to the point
- θ is the angle between the positive x-axis and the line segment from the origin to the point
- Useful for representing positions in circular or radial patterns
- extend polar coordinates to 3D space by adding a second angle (φ)
- r is the distance from the origin to the point
- θ is the azimuthal angle in the x-y plane from the x-axis (longitude)
- φ is the polar angle from the z-axis (colatitude or zenith angle)
- Commonly used in geospatial applications and for representing positions on a sphere ($Earth, celestial coordinates)
Transformations
Translation, Rotation, and Scaling
- moves an object in 3D space by adding a vector (tx, ty, tz) to its coordinates
- Preserves the object's orientation and size
- Transformation matrix for translation: 1 & 0 & 0 & tx \\ 0 & 1 & 0 & ty \\ 0 & 0 & 1 & tz \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
- turns an object around an axis by a specified angle
- Rotation matrices for the x, y, and z axes:
- Rotation around the x-axis by angle α: 1 & 0 & 0 & 0 \\ 0 & \cos\alpha & -\sin\alpha & 0 \\ 0 & \sin\alpha & \cos\alpha & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
- Rotation around the y-axis by angle β: \cos\beta & 0 & \sin\beta & 0 \\ 0 & 1 & 0 & 0 \\ -\sin\beta & 0 & \cos\beta & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
- Rotation around the z-axis by angle γ: \cos\gamma & -\sin\gamma & 0 & 0 \\ \sin\gamma & \cos\gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
- Rotation matrices for the x, y, and z axes:
- changes the size of an object by multiplying its coordinates with a scaling factor (sx, sy, sz)
- Uniform scaling applies the same factor to all dimensions ($sx = sy = sz)
- Non-uniform scaling allows for different factors along each axis
- Transformation matrix for scaling: sx & 0 & 0 & 0 \\ 0 & sy & 0 & 0 \\ 0 & 0 & sz & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}$$
Transformation Matrices and Composition
- Transformation matrices are 4x4 matrices that encode translation, rotation, and scaling operations
- Allows for applying multiple transformations to an object using matrix multiplication
- The order of matrix multiplication matters, as transformations are not commutative
- Composition of transformations involves multiplying transformation matrices in a specific order
- Example: To scale, rotate, and then translate an object, multiply the matrices in the following order:
- Changing the order of transformations can result in different final positions and orientations
Rotation Representations
Euler Angles and Gimbal Lock
- Euler angles represent rotations as a sequence of three rotations around the principal axes (x, y, z)
- Common conventions include XYZ, ZYX, and YZX (roll, pitch, yaw)
- Each rotation is represented by an angle (α, β, γ)
- Euler angles are intuitive and widely used in robotics and aviation
- Gimbal lock is a problem that occurs when two of the three rotation axes align, resulting in a loss of one degree of freedom
- Happens when the second rotation (e.g., pitch) is ±90 degrees
- Can cause unexpected behavior and singularities in calculations
- Avoiding gimbal lock requires careful planning of rotation sequences or using alternative representations
Quaternions and Advantages
- Quaternions are a 4D extension of complex numbers, consisting of a scalar part (w) and a vector part (x, y, z)
- Represent rotations in 3D space without suffering from gimbal lock
- multiplication is used to compose rotations, similar to matrix multiplication
- Quaternions are more compact and numerically stable compared to rotation matrices
- Advantages of using quaternions for rotations:
- Avoid gimbal lock, as quaternions represent rotations uniquely and continuously
- Interpolation between rotations (slerp) is more straightforward and computationally efficient
- Quaternion multiplication is faster than matrix multiplication for composing rotations
- Normalized quaternions (unit quaternions) ensure valid rotations and simplify calculations
Key Terms to Review (20)
Affine transformation: An affine transformation is a mathematical operation that transforms points in a coordinate space through linear mapping and translation. This transformation preserves points, straight lines, and planes, meaning that parallel lines remain parallel after the transformation. In the context of 3D coordinate systems and transformations, affine transformations are essential for manipulating objects, including scaling, rotation, translation, and shearing without altering their fundamental structure.
Bézier curves: Bézier curves are mathematical curves that are widely used in computer graphics and related fields for modeling smooth curves that can be scaled indefinitely. They are defined by a set of control points, which determine the shape and direction of the curve, making them essential for tasks such as rendering shapes, animations, and motion paths. Understanding Bézier curves helps in manipulating 3D geometry and transformations, managing 3D coordinate systems, and optimizing CPU and GPU performance.
Cartesian Coordinates: Cartesian coordinates are a system used to define the position of points in a multi-dimensional space using a set of numerical values. This system is built on perpendicular axes that create a grid, allowing for precise representation of locations in 2D or 3D space, which is crucial for spatial modeling and transformations.
Catmull-Rom splines: Catmull-Rom splines are a type of cubic spline used in computer graphics for creating smooth curves that pass through a set of given points. These splines are particularly valuable because they maintain continuity and provide a good level of control over the curve's shape, making them ideal for tasks such as animation paths and 3D modeling.
Composability: Composability refers to the ability to combine different elements or systems in a flexible and modular manner to create new functionalities or experiences. This concept is crucial in 3D coordinate systems and transformations, as it allows developers to build complex scenes and interactions by reusing existing components while maintaining consistency and coherence in transformations across multiple objects.
Coordinate transformation: Coordinate transformation refers to the mathematical process of converting the representation of points or vectors from one coordinate system to another. This concept is crucial in fields like computer graphics and augmented reality, as it enables the accurate representation and manipulation of objects in 3D space, ensuring that their positions and orientations can be consistently understood across different coordinate frameworks.
Homogeneous coordinates: Homogeneous coordinates are an extension of traditional Cartesian coordinates used in projective geometry, where an extra dimension is introduced to facilitate the representation of points at infinity and to simplify the mathematics of transformations. This system allows for easier computations when performing operations such as translation, scaling, and rotation, as it enables these transformations to be expressed as matrix multiplications. By using homogeneous coordinates, a point in 2D space represented as (x, y) can be expressed as (wx, wy, w), where w is a non-zero scaling factor.
Homogeneous transformation: A homogeneous transformation is a mathematical operation that combines translation, rotation, and scaling in a single matrix representation, facilitating the manipulation of 3D objects in computer graphics and robotics. This transformation uses a 4x4 matrix to represent both the position and orientation of an object in a unified manner, making it easier to perform multiple transformations sequentially. By employing homogeneous coordinates, it allows for operations like perspective projection and simplifies the calculations involved in transforming points in 3D space.
Invertibility: Invertibility refers to the property of a transformation that allows it to be reversed, meaning there exists an inverse transformation that can restore the original state. This concept is crucial in the context of 3D coordinate systems and transformations, as it ensures that operations such as translations, rotations, and scaling can be undone or reverted back to their original configuration. Understanding invertibility allows for the manipulation of 3D objects while maintaining the integrity of their positioning and orientation in space.
Local vs. Global Coordinates: Local vs. global coordinates refer to two different frameworks used to define positions and transformations in a three-dimensional space. Local coordinates are specific to an object or a local reference frame, meaning that they describe the object's position and orientation relative to its own origin. Global coordinates, on the other hand, are defined with respect to a fixed reference frame that applies universally across the entire environment, allowing for consistent positioning and movement of objects in relation to each other.
Matrix representation: Matrix representation refers to the use of matrices to mathematically describe geometric transformations in 3D space, such as translation, rotation, and scaling. This concept is crucial for working with 3D coordinate systems, as it provides a compact and efficient way to manipulate points and shapes in virtual environments. By using matrix multiplication, complex transformations can be combined into a single operation, greatly simplifying calculations and enhancing performance in augmented and virtual reality applications.
Mesh: A mesh is a collection of vertices, edges, and faces that defines the shape of a 3D object in space. It serves as the backbone for 3D models used in various applications, enabling transformations and manipulations that define how the object behaves and appears within different coordinate systems. By using meshes, we can create detailed and complex geometries that enhance the realism and interactivity of virtual environments.
OpenGL: OpenGL is an open-source graphics library that provides a cross-language, cross-platform API for rendering 2D and 3D vector graphics. It allows developers to interact with a computer's graphics hardware to create high-performance visual applications, making it essential for graphics programming in augmented and virtual reality contexts. Its capability to handle 3D coordinate systems and transformations plays a crucial role in displaying realistic environments, while its historical milestones mark significant advancements in AR/VR hardware and software development.
Polygonal model: A polygonal model is a digital representation of a 3D object using polygons, typically triangles or quadrilaterals, as the building blocks. These models are fundamental in 3D graphics and computer animation, serving as the basis for rendering visual representations in various applications, including video games and virtual environments. The way these polygons are arranged and transformed within 3D coordinate systems directly influences the object's appearance and behavior in a virtual space.
Quaternion: A quaternion is a mathematical construct that extends complex numbers to higher dimensions and is used to represent rotations in three-dimensional space. Quaternions consist of one real part and three imaginary parts, allowing for smooth and efficient interpolation of rotations without suffering from gimbal lock, a common issue when using Euler angles. This property makes quaternions particularly useful in computer graphics, robotics, and aerospace applications where 3D transformations are critical.
Rotation: Rotation refers to the circular movement of an object around a central point or axis. In 3D space, rotation is a fundamental transformation that changes the orientation of an object while keeping its shape and size intact. This concept is essential for manipulating objects in virtual environments, enabling realistic movements and interactions within both augmented and virtual reality settings.
Scaling: Scaling refers to the process of resizing objects in a virtual environment, altering their dimensions in relation to a reference point. It plays a crucial role in 3D geometry and transformations by enabling objects to be enlarged or reduced without distorting their proportions. In the context of 3D coordinate systems, scaling affects how objects interact with the coordinate space, impacting their visual representation and spatial relationships within that environment.
Spherical coordinates: Spherical coordinates are a three-dimensional coordinate system that represents points in space using three values: the radial distance from a reference point, the polar angle measured from a vertical axis, and the azimuthal angle measured in the horizontal plane. This system is particularly useful for dealing with problems involving symmetry, such as those found in physics and engineering. By converting from Cartesian coordinates, which use x, y, and z axes, spherical coordinates simplify calculations related to spheres and circular motion.
Translation: Translation refers to the process of moving an object from one location to another in a three-dimensional space, typically defined by its coordinates. It involves altering the position of an object without changing its orientation, scale, or shape. This transformation is crucial in 3D geometry as it allows for the manipulation of objects in virtual environments and is essential for accurately representing spatial relationships within 3D coordinate systems.
Unity 3D: Unity 3D is a cross-platform game engine developed by Unity Technologies, widely used for creating both 2D and 3D games and applications. Its flexibility allows developers to build immersive virtual and augmented reality experiences, making it a key tool in the AR/VR development landscape. The engine supports a variety of 3D coordinate systems and transformations, enabling the precise placement and manipulation of objects in virtual spaces.