UBI 516 Advanced Computer Graphics Three Dimensional Viewing Aydın Öztürk ozturk@ube.ege.edu.tr http://www.ube.ege.edu.tr/~ozturk

Overview Viewing a 3D scene Projections Parallel and perspective

Overview Depth cueing and hidden surfaces Identifying visible lines and surfaces

Overview Surface rendering

Overview Exploded and cutaway views

Overview 3D and stereoscopic viewing

3D Viewing Pipeline MC DC Modeling Transformation Viewport Transformation NC WC Normalization Transformation and Clipping Viewing Transformation VC PC Projectionn Transformation

Viewing Coordinates Generating a view of an object in 3D is similar to photographing the object. Whatever appears in the viewfinder is projected onto the flat film surface. Depending on the position, orientation and aperture size of the camera corresponding views of the scene is obtained.

Specifying The View Coordinates xw zw yw xv zv yv P0=(x0 , y0 , z0) For a particular view of a scene first we establish viewing-coordinate system. A view-plane (or projection plane) is set up perpendicular to the viewing z-axis. World coordinates are transformed to viewing coordinates, then viewing coordinates are projected onto the view plane.

Specifying The View Coordinates To establish the viewing reference frame, we first pick a world coordinate position called the view reference point. This point is the origin of our viewing coordinate system. If we choose a point on an object we can think of this point as the position where we aim a camera to take a picture of the object.

Specifying The View Coordinates Next, we select the positive direction for the viewing z-axis, and the orientation of the view plane, by specifying the view-plane normal vector, N. We choose a world coordinate position P and this point establishes the direction for N. OpenGL establishes the direction for N using the point P as a look at point relative to the viewing coordinate origin. yv xv xv yw zv N P0 P xw zw

Specifying The View Coordinates Finally, we choose the up direction for the view by specifying view-up vector V. This vector is used to establish the positive direction for the yv axis. The vector V is perpendicular to N. Using N and V, we can compute a third vector U, perpendicular to both N and V, to define the direction for the xv axis. yv xv V yw zv N P0 P xw zw

Specifying The View Coordinates To obtain a series of views of a scene , we can keep the the view reference point fixed and change the direcion of N. This corresponds to generating views as we move around the viewing coordinate origin. P0 N N

Transformation From World To Viewing Coordinates Conversion of object descriptions from world to viewing coordinates is equivalent to transformation that superimpoes the viewing reference frame onto the world frame using the translation and rotation. xv yv zv yw xw zw

Transformation From World To Viewing Coordinates First, we translate the view reference point to the origin of the world coordinate system xv yv zv yw xw zw

Transformation From World To Viewing Coordinates Second, we apply rotations to align the xv,, yv and zv axes with the world xw, yw and zw axes, respectively. yw yv zv xv yv zv xv xw zw

Transformation From World To Viewing Coordinates If the view reference point is specified at word position (x0, y0, z0), this point is translated to the world origin with the translation matrix T.

Transformation From World To Viewing Coordinates The rotation sequence requires 3 coordinate-axis transformation depending on the direction of N. First we rotate around xw-axis to bring zv into the xw -zw plane.

Transformation From World To Viewing Coordinates Then, we rotate around the world yw axis to align the zw and zv axes.

Transformation From World To Viewing Coordinates The final rotation is about the world zw axis to align the yw and yv axes.

Transformation From World To Viewing Coordinates The complete transformation from world to viewing coordinate transformation matrix is obtaine as the matrix product

Transformation From World To Viewing Coordinates Another method for generating the rotation-transformation matrix is to calculate uvn vectors and obtain the composite rotation matrix directly. Given the vectors and , these unit vectors are calculated as

Transformation From World To Viewing Coordinates This method also automatically adjusts the direction for so that is perpendicular to . The rotation matrix for the viewing transformation is then

Transformation From World To Viewing Coordinates The matrix for translating the viewing origin to the world origin is

Transformation From World To Viewing Coordinates The composite matrix for the viewing transformation is then

Transformation From World To Viewing Coordinates: An Example For 2d System 0 2 4 6 Θ=300 P0=(4,3) 0 2 4 6 x

Transformation From World To Viewing Coordinates: An Example For 2d System Translation: y 0 2 4 6 P 2 x′ 2 y′ 2 4 6 Θ=300 P0 x

Transformation From World To Viewing Coordinates: An Example For 2d System Rotation y x x′ y′ 2 4 6 P P0 0 2 4 6

Transformation From World To Viewing Coordinates: An Example For 2d System New coordinates

Transformation From World To Viewing Coordinates: An Example For 2d System Alternative Method y 0 1 2 3 P 1 x′ y′ 1 n v Θ=300 P0 1 2 3 x

Projections Once WC description of the objects in a scene are converted to VC we can project the 3D objects onto 2D view-plane. Two types of projections: -Parallel Projection -Perspective Projection

Classical Viewings Hand drawings : Determined by a specific relationship between the object and the viewer.

Parallel Projections Coordinate Positions are transformed to the view plane along parallel lines. View Plane P2 P′2 P1 P′1

Parallel Projections Orthographic parallel projection The projection is perpendicular to the view plane. Oblique parallel projecion The parallel projection is not perpendicular to the view plane.

Orthographic Parallel Projection The orthographic transformation

Orthographic Parallel Projection

Oblique Parallel Projection The projectors are still ortogonal to the projection plane But the projection plane can have any orientation with respect to the object. It is used extensively in architectural and mechanical design.

Oblique Parallel Projection Preserve parallel lines but not angles Isometric view : Projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of object. Dimetric view : Symmetric with two faces. Trimetric view : General case.

Oblique Parallel Projection Preserve parallel lines but not angles Isometric view : Projection plane is placed symmetrically with respect to the three principal faces that meet at a corner of object. Dimetric view : Symmetric with two faces. Trimetric view : General case.

Oblique Parallel Projection yv (xp, yp) α (x, y, z) L φ xv (x, y) zv

Oblique Parallel Projection The oblique transformation

Oblique Parallel Projection

Perspective Projections First discovered by Donatello, Brunelleschi, and DaVinci during Renaissance Objects closer to viewer look larger Parallel lines appear to converge to single point

Perspective Projections In perspective projection object positions are transformed to the view plane along lines that converge to a point called the projection reference point (or center of projection)

Perspective Projections In the real world, objects exhibit perspective foreshortening: distant objects appear smaller The basic situation:

Perspective Projections When we do 3-D graphics, we think of the screen as a 2-D window onto the 3-D world: How tall should this bunny be?

Perspective Projections The geometry of the situation is that of similar triangles. View from above: P (x, y, z) X Z (0,0,0) x′ = ? View plane (xp, yp) d

Perspective Projections Desired result for a point [x, y, z, 1]T projected onto the view plane:

Perspective Projections

Perspective Projections

Projection Matrix We talked about geometric transforms, focusing on modeling transforms Ex: translation, rotation, scale, gluLookAt() These are encapsulated in the OpenGL modelview matrix Can also express projection as a matrix These are encapsulated in the OpenGL projection matrix

View Volumes When a camera used to take a picture, the type of lens used determines how much of the scene is caught on the film. In 3D viewing, a rectangular view window in the view plane is used to the same effect. Edges of the view window are parallel to the xv-yv axes and window boundary positions are specified in viewing coordinates.

View Volumes Parallel Projection Parallel Projection View volume (frustum) window zv Back Plane window Front Plane Back Plane Projection Reference Point Front Plane Parallel Projection Parallel Projection Perspective Projection

Clipping An algorithm for 3D clipping identifies and saves all surface segments within the view volume for display. All parts of object that are outside the view volume are discarded.

Clipping Lines To clip a line against the view volume, we need to test the relative position of the line using the view volume’s boundary plane equation. An end point (x,y,z) of a line segment is outside a boundary plane if where A, B, C and D are the plane parameters for that boundary.

Clipping Polygon Surface To clip a polygon surface, we can clip the individual polygon edges. First we test the coordinate extends against each boundary of the view volume to determine whether the object is completely inside or completely outside of that boundary. If the object has intersection with the boundary then we apply intersection calculations.

Clipping Polygon Surface The projection operation can take place before the view- volume clipping or after clipping. All objects within the view volume map to the interior of the specified projection window. The last step is to transform the window contents to a 2D view port.

Clipping Polygon Surface Viev volume

Steps For Normalized View Volumes A scene is constructed by transforming object descriptions from modeling coordinates to wc. The world descriptions are converted to viewing coordinates. The viewing coordinates are transformed to projection coordinates which effectively converts the view volume into a rectangular parallelepiped. The parallelepiped is mapped into the unit cube called normalized projection coordinate system. A 3D viewport within the unit cube is constructed. Normalized projection coordinates are converted to device coordinates for display.

Normalized View Volumes y x (Xwmax, ywmax, zback) (Xvmax, yvmax, zvmax) z (Xwmin, ywmin, zfront) Parallelepiped View Volume (Xvmin, yvmin, zwmin) Unit Cube

Orthogonal Projection Normalization

Oblique Projection Normalization Angles of projection  for x axis  for y axis Shearing matrix H(, )  

Oblique Projection Normalization Finished ? No, this is a sheared view volume, so we have to apply orthogonal transformation : P=Porth STH

Perspective Projection Normalization Perspective Normalization is Trickier

Perspective Projection Normalization Consider N = After multiplying: p’ = Np

Perspective Projection Normalization After dividing by w’, p’ -> p’’

Perspective Projection Normalization Quick Check If x = z x’’ = -1 If x = -z x’’ = 1

Perspective Projection Normalization What about z? if z = zmax if z = zmin Solve for a and b such that zmin -> -1 and zmax ->1 Resulting z’’ is nonlinear, but preserves ordering of points If z1 < z2 … z’’1 < z’’2

Perspective Projection Normalization We did it. Using matrix, N Perspective viewing frustum transformed to cube Orthographic rendering of cube produces same image as perspective rendering of original frustum

OpenGL Projection Commands

OpenGL Look-At Function OpenGL utility function VRP: eyePoint (eyex, eyey, eyez) VPN: – ( atPoint – eyePoint ) (atx, aty, atz) – (eyex, eyey, eyez) VUP: upPoint – eyePoint (upx, upy, upz) gluLookAt(eyex, eyey, eyez, atx, aty, atz, upx, upy, upz); look-at positioning

Projections in OpenGL Angle of view, field of view : Only objects that fit within the angle of view of the camera appear in the image View volume, view frustum : Be clipped out of scene Frustum – truncated pyramid

Projections in OpenGL

Perspective in OpenGL Specification of a frustum near, far: positive number !!  zmax = – far  zmin = – near glMatrixMode(GL_PROJECTION); glLoadIdentity( ); glFrustum(xmin, xmax, ymin, ymax, near, far);

Perspective in OpenGL Specification using the field of view fov: angle between top and bottom planes fovy: the angle of view in the up (y) direction aspect ratio: width / height glMatrixMode(GL_PROJECTION); glLoadIdentity( ); gluPerspective(fovy, aspect, near, far);

Parallel Viewing in OpenGL Orthographic viewing function OpenGL provides only this parallel-viewing function near < far !!  no restriction on the sign  zmax = – far  zmin = – near glMatrixMode(GL_PROJECTION); glLoadIdentity( ); glOrtho(xmin, xmax, ymin, ymax, near, far);

Optional Clipping Planes glClipPlane(id, PlaneParameters); glEnable(id); // id = GL_CLIP_PLANE0, GL_CLIP_PLANE1, ... // PlaneParameters = A,B,C and D of the plane . . . glDisable(id);