COMP Computer Graphics and Image Processing. a6: Projections. In part 2 of our study of Viewing, we ll look at. COMP27112 Toby Howard

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1 Computer Graphics and Image Processing a6: Projections Introduction In part 2 of our stud of Viewing, we ll look at The theor of geometrical planar projections Classes of projections Perspective Parallel Basic mathematics of projections Matrix representations of projections Temple figures courtes: Prof. Ed Angel, Universit of New Mexico 2

2 3D Viewing: the camera analog Real World Computer Graphics Arrange the scene into the desired composition Set Modelling Transformation 2 Point the camera at the scene Set Viewing Transformation 3 Choose the camera lens, or adjust the zoom Set Projection Transformation 4 Determine the size and shape of the final photograph Set Viewport Transformation 3 The 3D Viewing Pipeline 3D vertex x z Modelling transformation M 2 3 Viewing transformation V Note: In OpenGL, M and V are combined into a single modelview matrix (see later) Projection transformation P 4 Clip to view volume Perspective division 5 6 Viewport transformation 2D pixel px,p In the last lecture we covered steps and 2. In this lecture, we ll look at steps

3 Plamar geometric projections How do we specif the mapping from 3D coordinates to 2D coordinates on the screen? We clearl have to somehow convert all the 3D coordinates into suitable 2D coordinates. This conversion process is called projection. There are two classes of planar geometric projection: Parallel projection Perspective projection Note: We re considering onl planar geometric projections: the kind that project lines to lines. Other kinds of projections change lines into curves and vice versa (used in cartograph, for example), which we will not stud. 3D 2D 5 Planar Geometric Projections Parallel Perspective Orthographic Top Front Oblique Cavalier Cabinet -point 2-point 3-point Side Axonometric Isometric Dimetric Trimetric In Computer Graphics, perspective projections are common, for their sense of realism. Parallel projections are often used in CAD and Engineering Drawing, because the allow precise measurements to be taken from the image. 6 3

4 Parallel projection The projection or image is the set of points at which the projectors intersect the projection plane. Parallel edges remain parallel in the projection. Angles between edges ma be distorted. Projectors 3D object 2D projection plane, oriented in 3D space Centre of projection, or ee point, is at infinit, so projectors are parallel 7 Parallel projection 3D object Projectors Projectors 2D projection plane, oriented in 3D space 8 4

5 Parallel projection: orthographic In orthographic parallel projection, projectors are perpendicular to the projection plane The projection plane is parallel to one of the (X,Y,Z) axes of the 3D object being viewed The projected image shows onl 2 of the 3 axes There is no distortion of lengths or angles, so well-suited for engineering drawing and CAD Front Side Top 9 Parallel projection 3D object Projectors Projectors 2D projection plane, oriented in 3D space 0 5

6 Computing orthographic projection Suppose we have a projection plane located at z=0, and the plane is parallel to the XY plane A 3D point (x,,z) will project to (x p, p,z p) on the projection plane We want to find (x p, p,z p) Y Projector orthogonal to the XY plane (i.e., parallel to the Z-axis B definition: xp = x p = z = 0 p Projection plane, at z=0, parallel to the XY plane (x p, p,z p) (x,,z) X Z 2 6

7 Matrix for orthographic projection We can express the projection as the matrix xp x p = zp z Y Projector orthogonal to the XY plane (i.e., parallel to the Z-axis This is the same as scaling b (,,0) This matrix is singular We can derive similar matrices for other positions of the projection plane Projection plane, at z=0, parallel to the XY plane Z (x p, p,z p) (x,,z) X 3 Parallel projection: axonometric In axonometric parallel projection, projectors are perpendicular to the projection plane The projection plane can have an orientation relative to the object being viewed Now we can see the three axes. There is distortion of lengths or angles, but measurements can still be made Isometric: projection plane is smmetrical to (X,Y,Z) Dimetric: projection plane is smmetrical to 2 of (X,Y,Z) Trimetric: projection plane placed arbitraril 4 7

8 Parallel projection: oblique This is the most general case of parallel projection Projectors can make an angle with the projection plane The projection plane can have an orientation relative to the object being viewed We see the three axes. There is distortion of lengths or angles, but measurements can still be made We can of course derive matrices to perform all the parallel projections we have seen (but we will not go into the details in this course) 5 Planar Geometric Projections Parallel Perspective Orthographic Top Front Oblique Cavalier Cabinet -point 2-point 3-point Side Axonometric Isometric Dimetric Trimetric In Computer Graphics, perspective projections are common, for their sense of realism. Parallel projections are often used in CAD and Engineering Drawing, because the allow precise measurements to be taken from the image. 6 8

9 Perspective During the Renaissance, artists began to understand and use perspective This is an engraving b Albrecht Dürer, St Jerome dans sa Cellule (54) 7 Perspective machines Dürer and others used devices to help them draw with correct perspective 8 9

10 Perspective projection Perspective projection reflects the wa we see (ee s lens and retina) 3D object 2D projection plane, oriented in 3D space Centre of projection, or ee point 9 Perspective projection The projection or image is the set of points at which the projectors intersect the projection plane. 3D object Projectors 2D projection plane, oriented in 3D space Centre of projection, or ee point, is a point, so projectors converge 20 0

11 Perspective projection Objects further awa from the COP become smaller. Edges that were parallel ma converge. Angles between edges ma be distorted. 3D object 2D projection plane, oriented in 3D space 2 Small or far awa? With perspective projection, distant objects appear smaller in the projected image 22

12 Planar Geometric Projections Parallel Perspective Orthographic Top Front Oblique Cavalier Cabinet -point 2-point 3-point Side Axonometric Isometric Dimetric Trimetric In Computer Graphics, perspective projections are common, for their sense of realism. Parallel projections are often used in CAD and Engineering Drawing, because the allow precise measurements to be taken from the image. 23 Classical perspective projections How man of the (x,,z) axes of the model are parallel to the projection plane determines how man vanishing points are seen in the projected image 2 of (x,,z) are parallel to the projection plane: -point perspective of (x,,z) is parallel to the projection plane: 2-point perspective 0 of (x,,z) is parallel to the projection plane: 3-point perspective -point perspective 2-point perspective 3-point perspective 24 2

13 2-point perspective 25 Perspective machine projected point projector projection screen 3D object Albrecht Dürer 26 3

14 Computing perspective Suppose we have a projection plane located at z=d, and the plane is parallel to the XY plane A point (x,,z) will project to (x p, p,z p) on the projection plane We want to find (x p, p,z p) Y B definition, = d (x,,z) zp Projection plane, at z=d, parallel to the XY plane (x p, p,z p) X Z 27 Computing perspective We need to find xp and Looking at the top view, b similar triangles, we have: xp d x z dx z p x = xp = p x = z/d Z (x p,d) Top view (looking down Y axis) X z= d (x,z) Similarl, looking at the side view, we have: p = z/d Y ( p,d) (,z) Side view (looking down Z axis) X 28 4

15 Perspective projection matrix We can express this transformation as a 4x4 matrix: If we multipl a point b this matrix, we get the transformed point x z z/d Recall however that we must alwas end up with a 3D point which has w =, so we need to divide through all elements b w: This is called perspective division xʹ x ʹ = zʹ z w 0 0 / d 0 ʹ x z/d z/d z z/d x z/d z/d d 29 Perspective matrices: summar We derived the matrix to perform a -point projection where the projection plane is parallel to the XY plane. xʹ x ʹ = zʹ z w 0 0 /dz 0 ʹ We can also derive (details omitted) a matrix which expresses the most general case, 3-point projection, where the projection plane is not parallel to an of the XY, XZ or YZ planes: xʹ x ʹ = zʹ z w /dx /d /dz 0 ʹ We ve renamed d to d z just to remind us that it s a z-coordinate The projection plane intersects the X,Y and Z axes at (d x d d z ) 30 5

16 Projections: summar Parallel orthographic projection onto z=0 plane xp x p = zp z point perspective projection onto XY plane xʹ x ʹ = zʹ z w 0 0 / d 0 ʹ Then divide b wʹ to give 3D point x z/d z/d d 3 What can the camera see? What part of the world can a camera see? Onl objects in front of it Onl objects within its field of view, which depends on its lens (about 90 o for the human ee) Onl a finite distance (ver distant objects will project too small to be seen) Field of view 32 6

17 Viewing volumes and clipping So, having specified the position and orientation of the camera (see 3D Viewing () ) we now describe the field of view of the camera b defining a 3D view volume Onl those parts of 3D objects which lie inside the view volume are displaed all objects are clipped against the view volume We ll look at clipping shortl, but first we ll see how the view volumes are defined. 33 Defining the view volume In Viewing () we established a coordinate sstem for the camera with axes Ŝ Û ˆF We define a 3D view volume which is attached to the camera As we would expect, the shapes of the volumes for parallel and perspective projection are different. Û ˆF 3D view volume Ŝ 34 7

18 View volume for parallel projection A view volume is a 3D shape, defined b six planes For parallel orthographic projection, it s a cuboid The cuboid is defined b a near plane (the projection plane) and a far plane, and top/bottom, left/right planes The near and far planes are orthogonal to the camera s ˆF axis Û ˆF Ŝ far plane projection plane (near plane) 35 View volume for perspective projection For perspective projection, this view volume is a frustum, a truncated pramid The frustum is defined b a near plane (the projection plane) and a far plane The near and far planes are orthogonal to the camera s ˆF axis Û ˆF Ŝ far plane projection plane (near plane) 36 8

19 Orthographic projection in OpenGL projection plane (near plane) Y View volume: cuboid defined b 6 clip planes far plane top camera (ee) Z left right bottom near far X glortho (GLdouble left, GLdouble right, GLdouble top, GLdouble bottom, GLdouble near, GLdouble far); OpenGL 37 Perspective projection in OpenGL projection plane (near plane) Y W far plane FOV H Z camera (ee) near gluperspective (GLdouble fov, GLdouble aspect, GLdouble near, GLdouble far); X far OpenGL 38 9

20 Perspective projection in OpenGL Suppose the projection plane (near plane) is at z= -d, and the ee (or camera ) is at (0,0,0) Then the (-point) projection transformation, as we saw in Viewing (2) is given b / d 0 Z Y camera (ee) Note: this projection transformation is independent of the far plane which is onl for clipping d X 39 The clipping operation Clipping takes place in the cube produced b projection normalization (easier than clipping against a frustum) Clipping is easiest to visualise in 2D Here we see polgon being clipped against a rectangular clipping region clipping region clipping region There are established algorithms for clipping in 2D and 3D 40 20

21 Viewing volume and clipping tonchan s real photos 4 Perspective division The clipping operation returns a set of (x,,z,w) vertices defining polgons which are inside the view volume OpenGL now performs the perspective division b w to convert these (x,,z,w) values to (x,,z) 3D points Projection normalization Clip to view volume Perspective division New step, omitted from previous diagrams for clarit but we won t cover this 42 2

22 3D Viewing: the camera analog Real World Computer Graphics Arrange the scene into the desired composition Set Modelling Transformation 2 Point the camera at the scene Set Viewing Transformation 3 Choose the camera lens, or adjust the zoom Set Projection Transformation 4 Determine the size and shape of the final photograph Set Viewport Transformation 43 The 3D Viewing Pipeline 3D vertex x z Modelling transformation M 2 3 Viewing transformation V Note: In OpenGL, M and V are combined into a single modelview matrix (see later) Projection transformation P Clip to view volume 4 Perspective division 5 6 Viewport transformation 2D pixel px,p 44 22

23 The Viewport Transformation After perspective division, we now have 3D coordinates (-,) Y (,) OpenGL has scaled these into the range [-,+] in X, Y and Z In the final step, using onl the X and Y values, OpenGL computes a transformation from [-,+] to the displa screen (0,0) X Either the whole window Or a part of it called a viewport (-,-) (,-) But we retain Z, to remove hidden surfaces 45 Summar of 3D viewing We can now summarise the steps of the viewing process. The modelling transformation arranges objects in our 3D world 2. The viewing transformation transforms the world to give the same view as if it were being photographed b a camera 3. The projection transformation performs a parallel/perspective projection within limits (the clip planes) 4. Those parts of the 3D world outside the clip planes are discarded 5. If it s a perspective view, the perspective division flattens the image 6. The viewport transformation maps the final image to a position in part of the displa screen window

24 Julian Beever