3D projection - Citizendia

Views
Graphical projections Planar projection Perspective projection Parallel projection Orthographic projection Multiviews Plan, or floor plan Section Elevation Auxiliary view Axonometric projection Isometric projection Dimetric projection Trimetric projection Oblique projection Cavalier perspective Cabinet projection 3D projection Fisheye Stereoscopic Projection Anamorphic projection Map projection
Other views Bird's-eye view/Aerial view Top-down perspective Worm's-eye view
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3D Projection is a method of mapping three dimensional points to a two dimensional plane. Graphical projection is a Protocol by which an image of an imaginary three-dimensional object is projected onto a planar surface without the aid of mathematical calculation Planar projections are the subset of 3D graphical projections constructed by linearly mapping points in three dimensional space to points on a two-dimensional projection plane Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D A plan is an Orthographic projection of a 3-dimensional object from the position of a horizontal plane through the object A floor plan ( floorplan) in Architecture and Building engineering is a Diagram, usually to scale, of the relationships between rooms In Geometry, a cross section is the intersection of a body in 2-dimensional space with a line or of a body in 3-dimensional space with a plane etc An elevation is an Orthographic projection of a 3-dimensional object from the position of a horizontal plane beside an object An auxiliary view is an angle at which one can view an object that is not one of the primary views for an Orthographic projection. Axonometric projection ("to measure along axes" is a technique used in orthographic pictorials Isometric projection is a form of Graphical projection —more specifically an Axonometric projection. Dimetric projection is a form of Axonometric projection, in which its direction of viewing is such that two of the three axes of space appear equally foreshortened of which Trimetric projection is a form of Axonometric projection, where the direction of viewing is such that all of the three axes of space appear unequally foreshortened This article discusses imaging of three-dimensional objects For an abstract mathematical discussion see Projection (linear algebra. The cavalier perspective, also called cavalier projection or high view point, is a way to represent a three dimensional object on a flat drawing and more specifically Cabinet projection or sometimes cabinet perspective is a type of Oblique projection. Owned by Atlassian Software Systems, FishEye is a Revision control browser and search engine Stereoscopy, stereoscopic imaging or 3-D (three-dimensional imaging is any technique capable of recording three-dimensional visual Anamorphosis is a distorted projection or perspective requiring the viewer to use special devices or occupy a specific vantage point to reconstitute the image A map projection is any method of representing the Surface of a sphere or other shape on a plane. A bird's-eye view is a View of an object from above as though the observer were a Bird, often used in the making of Blueprints, Floor plans Top-down perspective, also sometimes referred to as bird's-eye view, overhead view or helicopter view, A worm's-eye view is a View of an object from below as though the observer were a Worm. As most current methods for displaying graphical data are based on planar two dimensional media, the use of this type of projection is widespread, especially in computer graphics, engineering and drafting.

1 Orthographic projection
2 Perspective projection
3 Diagram
4 See also
5 References
6 Further Reading

Orthographic projection

Orthographic projections are a small set of transforms often used to show profile, detail or precise measurements of a three dimensional object. Orthographic projection is a means of representing a three- Dimensional (3D object in two dimensions (2D Common names for orthographic projections include plan, cross-section, bird's-eye, and elevation.

The normal of the viewing plane (the camera direction) is always parallel to one of the 3D axes making the mathematical transformation very simple. To project the 3D point $a x$ , $a y$ , $a z$ onto the 2D point $b x$ , $b y$ using an Orthographic projection parallel to the y axis(Profile view), the following equation can be used:

b x = s x a x + c x

b y = s z a z + c z

Where the vector s is an arbitrary scale factor, and c is an arbitrary offset. These constants are optional, and can be used to properly align the viewport. The projection can be shown using Matrix notation (introducing a temporary vector d for clarity)

$\begin{bmatrix} {d_x } \\ {d_y } \\ \end{bmatrix} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\begin{bmatrix} {a_x } \\ {a_y } \\ {a_z } \\ \end{bmatrix}$

$\begin{bmatrix} {b_x } \\ {b_y } \\ \end{bmatrix} = \begin{bmatrix} {s_x } & 0 \\ 0 & {s_z } \\ \end{bmatrix}\begin{bmatrix} {d_x } \\ {d_y } \\ \end{bmatrix} + \begin{bmatrix} {c_x } \\ {c_z } \\ \end{bmatrix}$

While orthographically projected images represent the three dimensional nature of the object projected, they do not represent the object as it would recorded photographically or perceived by a viewer observing it directly. In particular, lengths at all points in an orthographically projected image are of the same scale regardless of whether they are far away or near to the virtual viewer. As a result, lengths near to the viewer appear foreshortened.

Perspective projection

See also Transformation matrix

The perspective projection requires greater definition. In Linear algebra, Linear transformations can be represented by matrices. A conceptual aid to understanding the mechanics of this projection involves treating the 2D projection as being viewed through a camera viewfinder. The camera's position, orientation, and field of view control the behaviour of the projection transformation. The field of view (also field of vision) is the angular extent of the observable world that is seen at any given moment The following variables are defined to describe this transformation:

$\mathbf{a}_{x,y,z}$ - the point in 3D space that is to be projected.
$\mathbf{c}_{x,y,z}$ - the location of the camera.
$\mathbf{\theta}_{x,y,z}$ - The rotation of the camera. When $\mathbf{c}_{x,y,z}$ =<0,0,0>, and $\mathbf{\theta}_{x,y,z}$ =<0,0,0>, the 3D vector <1,2,0> is projected to the 2D vector <1,2>.
$\mathbf{e}_{x,y,z}$ - the viewer position in camera space. ^[1]

Which results in:

$\mathbf{b}_{x,y}$ - the 2D projection of $\mathbf{a}$ .

First, we define a point $\mathbf{d}_{x,y,z}$ as a translation of point $\mathbf{a}$ into a coordinate system defined by $\mathbf{c}$ . In Euclidean geometry, a translation is moving every point a constant distance in a specified direction In Mathematics and its applications a coordinate system is a system for assigning an n - Tuple of Numbers or scalars to each point This is achieved by subtracting $\mathbf{c}$ from $\mathbf{a}$ and then applying a vector rotation matrix using $-\mathbf{\theta}$ to the result. In Mathematics, matrix addition is the operation of adding two matrices by adding the corresponding entries together In Matrix theory, a rotation matrix is a real Square matrix whose Transpose is its inverse and whose Determinant is +1 This transformation is often called a camera transform: ^[2] ^[3]

$\begin{bmatrix} \mathbf{d}_x \\ \mathbf{d}_y \\ \mathbf{d}_z \\ \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 \\ 0 & {\cos -\mathbf{\theta}_x } & {\sin -\mathbf{\theta}_x } \\ 0 & { - \sin -\mathbf{\theta}_x } & {\cos -\mathbf{\theta}_x } \\ \end{bmatrix}\begin{bmatrix} {\cos -\mathbf{\theta}_y } & 0 & { - \sin -\mathbf{\theta}_y } \\ 0 & 1 & 0 \\ {\sin -\mathbf{\theta}_y } & 0 & {\cos -\mathbf{\theta}_y } \\ \end{bmatrix}\begin{bmatrix} {\cos -\mathbf{\theta}_z } & {\sin -\mathbf{\theta}_z } & 0 \\ { - \sin -\mathbf{\theta}_z } & {\cos -\mathbf{\theta}_z } & 0 \\ 0 & 0 & 1 \\ \end{bmatrix}\left( {\begin{bmatrix} \mathbf{a}_x \\ \mathbf{a}_y \\ \mathbf{a}_z \\ \end{bmatrix} - \begin{bmatrix} \mathbf{c}_x \\ \mathbf{c}_y \\ \mathbf{c}_z \\ \end{bmatrix}} \right)$

Or, for those less comfortable with matrix multiplication:

$\begin{array}{lcl} d_x &= &\cos \theta_y\cdot(\sin \theta_z\cdot(a_y-c_y)+\cos \theta_z\cdot(a_x-c_x))-\sin \theta_y\cdot(a_z-c_z) \\ d_y &= &\sin \theta_x\cdot(\cos \theta_y\cdot(a_z-c_z)+\sin \theta_y\cdot(\sin \theta_z\cdot(a_y-c_y)+\cos \theta_z\cdot(a_x-c_x)))+\cos \theta_x\cdot(\cos \theta_z\cdot(a_y-c_y)-\sin \theta_z\cdot(a_x-c_x)) \\ d_z &= &\cos \theta_x\cdot(\cos \theta_y\cdot(a_z-c_z)+\sin \theta_y\cdot(\sin \theta_z\cdot(a_y-c_y)+\cos \theta_z\cdot(a_x-c_x)))-\sin \theta_x\cdot(\cos \theta_z\cdot(a_y-c_y)-\sin \theta_z\cdot(a_x-c_x)) \\ \end{array}$

This transformed point can then projected onto the 2D plane using the formula (here, x/y is used as the projection plane, literature also may use x/z):^[4]

$\begin{array}{lcl} \mathbf{b}_x &= &(\mathbf{d}_x - \mathbf{e}_x) (\mathbf{e}_z / \mathbf{d}_z) \\ \mathbf{b}_y &= &(\mathbf{d}_y - \mathbf{e}_y) (\mathbf{e}_z / \mathbf{d}_z) \\ \end{array}$

Or, in matrix form using homogeneous coordinates:

$\begin{bmatrix} \mathbf{f}_x \\ \mathbf{f}_y \\ \mathbf{f}_z \\ \mathbf{f}_w \\ \end{bmatrix}=\begin{bmatrix} 1 & 0 & 0 & -\mathbf{e}_x \\ 0 & 1 & 0 & -\mathbf{e}_y \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 1/\mathbf{e}_z & 0 \\ \end{bmatrix}\begin{bmatrix} \mathbf{d}_x \\ \mathbf{d}_y \\ \mathbf{d}_z \\ 1 \\ \end{bmatrix}$

and

$\begin{array}{lcl} \mathbf{b}_x &= &\mathbf{f}_x / \mathbf{f}_w \\ \mathbf{b}_y &= &\mathbf{f}_y / \mathbf{f}_w \\ \end{array}$

The distance of the camera plane to the viewer, $\mathbf{e}_z$ , directly relates to the field of view, where $\alpha=2 \cdot \tan^{-1}(1/\mathbf{e}_z)$ is the viewed angle. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations

Subsequent clipping and scaling operations may be necessary to map the 2D plane onto any particular display media.

Diagram

Image:Perspective Transform Diagram.png

To determine which screen x coordinate corresponds to a point at Ax,Az multiply the point coordinates by:

$\text{screen x coordinate}(Bx) = \text{model x coordinate}(Ax) \times \frac{\text{distance from eye to screen}(Bz)}{\text{distance from eye to point}(Az)}$

the same works for the screen y coordinate:

$\text{screen y coordinate}(By) = \text{model y coordinate}(Ay) \times \frac{\text{distance from eye to screen}(Bz)}{\text{distance from eye to point}(Az)}$

References

^ Ingrid Carlbom, Joseph Paciorek (Dec. Computer graphics are Graphics created by Computers and more generally the Representation and Manipulation of Pictorial Data 3D computer graphics (in contrast to 2D computer graphics) are graphics that use a three-dimensional representation of geometric data that is stored in the computer A video card, also known as a graphics accelerator card, display adapter, or graphics card, is a hardware component whose function is to Transform clipping and lighting ( T&L or sometimes TCL) is a term used in Computer graphics. Texture mapping is a method for adding detail surface texture or colour to a computer-generated graphic or 3D model. Perspective (from Latin perspicere to see through in the graphic arts such as drawing is an approximate representation on a flat surface (such as paper of an image as it is perceived Homography is a concept in the mathematical science of Geometry. In Mathematics, homogeneous coordinates, introduced by August Ferdinand Möbius in his 1827 work Der barycentrische Calcul, allow Affine transformations 1978). Planar Geometric Projections and Viewing Transformations v. 10 n. 4. ACM Computing Surveys (CSUR), p. The Association for Computing Machinery, or ACM, was founded in 1947 as the world's first scientific and educational Computing society 465-502. DOI:10.1145/356744.356750. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document.
^ Riley, K F (2006). Mathematical Methods for Physics and Engineering. Cambridge University Press, 931,942. Cambridge University Press (known colloquially as CUP is a Publisher given a Royal Charter by Henry VIII in 1534 DOI:10.2277/0521679710. A digital object identifier ( DOI) is a permanent identifier given to an Electronic document. ISBN 0521679710.
^ Goldstein, Herbert (1980). Classical Machanics 2nd Edn. , 146-148. ISBN 0201029189.
^ Sonka, M; Hlavac, V & Boyle, R (1995), Image Processing, Analysis & Machine Vision 2nd Edn. , Chapman and Hall, pp. 14, ISBN 0412455706

Contents

Orthographic projection

Perspective projection

Diagram

See also

References

Further Reading