CSCE 441 Computer Graphics 3-D Viewing Jinxiang Chai

Outline 3D Viewing Required readings: HB 10-1 to 10-10 Compile and run the codes in page 388 - opengl three-dimensional viewing program example 1

Taking Pictures Using A Real Camera Steps: - Identify interesting objects - Rotate and translate the camera to a desired camera viewpoint - Adjust camera settings such as focal length - Choose desired resolution and aspect ratio, etc. - Take a snapshot

Taking Pictures Using A Real Camera Steps: - Identify interesting objects - Rotate and translate the camera to a desired camera viewpoint - Adjust camera settings such as focal length - Choose desired resolution and aspect ratio, etc. - Take a snapshot Graphics does the same thing for rendering an image for 3D geometric objects

Rendering Images Using a Virtual Camera

3D Geometry Pipeline Object space World space View space Rotate and translate the camera Object space World space View space Focal length Aspect ratio & resolution Normalized projection space Screen/Image space 5

3D Geometry Pipeline Model space (Object space) Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations... Model space (Object space)

3D Geometry Pipeline World space (Object space) Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations... World space (Object space)

3D Geometry Pipeline Eye space (View space) Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations... Eye space (View space)

Normalized projection space 3D Geometry Pipeline Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations... Normalized projection space

Image space, window space, raster space, screen space, device space 3D Geometry Pipeline Before being turned into pixels by graphics hardware, a piece of geometry goes through a number of transformations... Image space, window space, raster space, screen space, device space

Normalized project space Taking Steps Together C M Object space World space View space P V Normalized project space Image space 11

OpenGL Codes 12

OpenGL Codes 13

OpenGL Codes 14

Normalized project space 3D Geometry Pipeline Object space World space View space Normalized project space Image space

Translate, scale &rotate 3D Geometry Pipeline Translate, scale &rotate Object space World space glTranslate*(tx,ty,tz)

Translate, scale &rotate 3D Geometry Pipeline Translate, scale &rotate Object space World space glScale*(sx,sy,sz)

Translate, scale &rotate 3D Geometry Pipeline Translate, scale &rotate Object space World space Rotate about r by the angle glRotate*

Normalized projection space 3D Geometry Pipeline Object space World space View space Normalized projection space Image space Screen space

3D Geometry Pipeline World space View space Now look at how we would compute the world->eye transformation World space View space

3D Geometry Pipeline World space View space Now look at how we would compute the world->eye transformation Rotate&translate World space View space

Camera Coordinate

Viewing Trans: gluLookAt gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz) 23

Camera Coordinate Canonical coordinate system - usually right handed (looking down –z axis) - convenient for project and clipping

Camera Coordinate Mapping from world to eye coordinates - eye position maps to origin - right vector maps to x axis - up vector maps to y axis - back vector maps to z axis

Viewing Transformation We have the camera in world coordinates We want to model transformation T which takes object from world to camera

Viewing Transformation We have the camera in world coordinates We want to model transformation T which takes object from world to camera Trick: find T-1 taking object from camera to world

Viewing Transformation We have the camera in world coordinates We want to model transformation T which takes object from world to camera Trick: find T-1 taking object from camera to world ?

Review: 3D Coordinate Trans. Transform object description from to p

Review: 3D Coordinate Trans. Transform object description from to p 30

Review: 3D Coordinate Trans. Transform object description from camera to world

Viewing Transformation Trick: find T-1 taking object from camera to world - eye position maps to origin - back vector maps to z axis - up vector maps to y axis - right vector maps to x axis

Viewing Transformation Trick: find T-1 taking object from camera to world H&B equation (10-4)

Viewing Trans: gluLookAt gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)

Viewing Trans: gluLookAt Mapping from world to eye coordinates gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz) How to determine ?

Viewing Trans: gluLookAt Mapping from world to eye coordinates gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz) How to determine ?

Viewing Trans: gluLookAt Mapping from world to eye coordinates gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)

Viewing Trans: gluLookAt Mapping from world to eye coordinates gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz)

Viewing Trans: gluLookAt Mapping from world to eye coordinates gluLookAt (eyex,eyey,eyez,atx,aty,atz,upx, upy,upz) Stop here. H&B equation (10-1)

Viewing Trans: gluLookAt Test: gluLookAt( 4.0, 2.0, 1.0, 2.0, 4.0, -3.0, 0, 1.0, 0 ) - What’s the transformation matrix from the world space to the camera reference system?

3D-3D viewing transformation 3D Geometry Pipeline 3D-3D viewing transformation World space View space

Projection General definition transform points in n-space to m-space (m<n) In computer graphics map 3D coordinates to 2D screen coordinates

Projection General definition transform points in n-space to m-space (m<n) In computer graphics map 3D coordinates to 2D screen coordinates How can we project 3d objects to 2d screen space?

How Do We See the World? Let’s design a camera: idea 1: put a piece of film in front of camera Do we get a reasonable picture?

Pin-hole Camera Add a barrier to block off most of the rays This reduces blurring The opening known as the aperture How does this transform the image?

Camera Obscura The first camera Known to Aristotle Depth of the room is the focal length Pencil of rays – all rays through a point

Perspective Projection Maps points onto “view plane” along projectors emanating from “center of projection” (COP)

Perspective Projection Maps points onto “view plane” along projectors emanating from “center of projection” (COP) What’s relationship between 3D points and projected 2D points? 48

3D->2D Consider the projection of a 3D point on the camera plane

3D->2D Consider the projection of a 3D point on the camera plane 50

3D->2D Consider the projection of a 3D point on the camera plane By similar triangles, we can compute how much the x and y coordinates are scaled

3D->2D Consider the projection of a 3D point on the camera plane By similar triangles, we can compute how much the x and y coordinates are scaled

Homogeneous Coordinates Is this a linear transformation?

Homogeneous Coordinates Is this a linear transformation? no—division by z is nonlinear

Homogeneous Coordinates Is this a linear transformation? no—division by z is nonlinear Trick: add one more coordinate: homogeneous image coordinates homogeneous scene coordinates

Homogeneous Point Revisited Remember how we said 2D/3D geometric transformations work with the last coordinate always set to one What happens if the coordinate is not one We divide all coordinates by w: If w=1, nothing happens Sometimes, we call this division step the “perspective divide”

The Perspective Matrix Now we can rewrite the perspective projection equation as matrix-vector multiplications

The Perspective Matrix Now we can rewrite the perspective projection equation as matrix-vector multiplications This becomes a linear transformation!

The Perspective Matrix Now we can rewrite the perspective projection equation as matrix-vector multiplications After the division by w, we have

Perspective Effects Distant object becomes small The distortion of items when viewed at an angle (spatial foreshortening)

Perspective Effects Distant object becomes small The distortion of items when viewed at an angle (spatial foreshortening)

Perspective Effects Distant object becomes small The distortion of items when viewed at an angle (spatial foreshortening)

Properties of Perspective Proj. Perspective projection is an example of projective transformation - lines maps to lines - parallel lines do not necessary remain parallel - ratios are not preserved

Properties of Perspective Proj. Perspective projection is an example of projective transformation - lines maps to lines - parallel lines do not necessary remain parallel - ratios are not preserved One of advantages of perspective projection is that size varies inversely proportional to the distance-looks realistic

Vanishing Points What happens to parallel lines they are not parallel to the projection plane?

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? The equation of the line:

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? The equation of the line: After perspective transformation, we have

Vanishing Points (cont.) Letting t go to infinity:

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? The equation of the line:

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? The equation of the line: How about the line

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? The equation of the line: How about the line

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? The equation of the line: How about the line Same vanishing point!

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? Each set of parallel lines intersect at a vanishing point on the PP

Vanishing Points What happens to parallel lines they are not parallel to the projection plane? Each set of parallel lines intersect at a vanishing point on the PP

Parallel Projection Center of projection is at infinity Direction of projection (DOP) same for all points

Orthographic Projection Direction of projection (DOP) perpendicular to view plane

Orthographic Projection Direction of projection (DOP) perpendicular to view plane

Properties of Parallel Projection Not realistic looking Good for exact measurement Are actually affine transformation - parallel lines remain parallel - ratios are preserved - angles are often not preserved Most often used in CAD, architectural drawings, etc. where taking exact measurement is important

3D->2D Perspective projection from 3D to 2D 79

3D->2D Perspective projection from 3D to 2D But so far, we have not considered the size of film plane! We have also not considered visibility problem 80

Normalized project space 3D Geometry Pipeline Object space World space View space z = -1 z = 1 Normalized project space Image space Screen space

Perspective Projection Volume The center of projection and the portion of projection plane that map to the final image form an infinite pyramid. The sides of pyramid called clipping planes Additional clipping planes are inserted to restrict the range of depths

Perspective Projection Volume The center of projection and the portion of projection plane that map to the final image form an infinite pyramid. The sides of pyramid called clipping planes Additional clipping planes are inserted to restrict the range of depths

Perspective Projection Volume

Perspective Projection Volume

Perspective Projection Volume

Perspective Projection Volume

OpenGL Perspective-Projection z = -1 z = 1 Normalized project space View space End here glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar) gluPerspective(fovy,aspect,dnear, dfar) 88

General Perspective-Projection zfar (xwmax,ywmax,znear) (xwmin,ywmin,Znear) View space glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar) Six parameters define six clipping planes!

General Perspective-Projection glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar) near clipping plane

General Perspective-Projection glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar) far clipping plane

General Perspective-Projection glFrustum(xwmin, xwmax,ywmin, ywmax ,dnear,dfar) The size of the near plane

General Perspective-Projection B’=(1,1,1) B A A’=(-1,-1,-1) H&B equation (10-40)

General Perspective-Projection B’=(1,1,1) B A A’=(-1,-1,-1) View space (right-handed) Normalized project space (left-handed) A maps to A’, B maps to B’ Keep the directions of x and y axes!

General Perspective-Projection B’=(1,1,1) B A A’=(-1,-1,-1) View space (right-handed) Normalized project space (left-handed) dnear=-1, dfar=1 xmin=-1,xmax=1,ymin=-1,ymax=1

OpenGL Symmetric Perspective-Projection Function gluPerspective(fovy,aspect,dnear, dfar) Assume z-axis is the centerline of 3D view frustum!

Normalized projection space 3D Geometry Pipeline Object space World space View space Normalized projection space Image space

Viewport Transformation Display window (xvmin,yvmin) Image space-.>Image space glViewport(xvmin, yvmin, width, height)

Viewport Transformation - Besides x and y, each pixel has a depth value z, which is stored in depth buffer. Depth values will be used for visibility testing The color of the pixel is stored in color buffer (xvmin,yvmin) Image space-.>Image space glViewport(xvmin, yvmin, width, height)

Viewport Transformation From normalized projection coordinates to 3D screen coordinates dnear= -1, dfar= 1 xmin= -1,xmax= 1 ymin= -1,ymax= 1 dnear= 0, dfar= 1 xmin= xvmin,xmax= xvmax, ymin= yvmin,ymax= yvmax glViewport(xvmin, yvmin, width, height) xmax=xvmin+width,ymax=yvmin+ height

Viewport Transformation 0<=z<=1 B’=(1,1,1) B’’=(xvmax,yvmax,1) -1<=z<=1 Normalized project space A’’=(xvmin,yvmin,0) A’=(-1,-1,-1) Normalized projection space 3D screen space

Viewport Transformation 0<=z<=1 B’=(1,1,1) B’’=(xvmax,yvmax,1) -1<=z<=1 Normalized project space A’’=(xvmin,yvmin,0) A’=(-1,-1,-1) Normalized projection space 3D screen space H&B equation (10-42)

Summary: 3D Geometry Pipeline Object space World space View space Normalized projection space Image space

Normalized project space Taking Steps Together C M Object space World space View space P V Normalized project space Image space

OpenGL Codes

OpenGL Codes 106

OpenGL Codes 107

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