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Three-Dimensional Viewing Sang Il Park Sejong University Lots of slides are stolen from Jehee Lee’s

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Viewing Pipeline

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Virtual Camera Model Viewing Transformation –The camera position and orientation is determined Projection Transformation –The selected view of a 3D scene is projected onto a view plane

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General 3D Viewing Pipeline Modeling coordinates (MC) World coordinates (WC) Viewing coordinates (VC) Projection coordinates (PC) Normalized coordinates (NC) Device coordinates (DC)

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Viewing-Coordinate Parameters View point (eye point or viewing position) View-plane normal vector N

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Viewing-Coordinate Parameters Look-at point P ref View-up vector V –N and V are specified in the world coordinates –V should be perpendicular to N, but it can be difficult to a direction for V that is precisely perpendicular to N

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Viewing-Coordinate Reference Frame The camera orientation is determined by the uvn reference frame u n v

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World-to-Viewing Transformation Transformation from world to viewing coordinates –Translate the viewing-coordinate origin to the world- coordinate origin –Apply rotations to align the u, v, n axes with the world x w, y w, z w axes, respectively u n v

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World-to-Viewing Transformation

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Perspective Projection Pin-hold camera model –Put the optical center (Center Of Projection) at the origin –Put the image plane (Projection Plane) in front of the COP –The camera looks down the negative z axis we need this if we want right-handed-coordinates

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Parallel Projection Special case of perspective projection –Distance from the COP to the PP is infinite –Also called “parallel projection” –What’s the projection matrix? Image World Slide by Steve Seitz

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Taxonomy of Geometric Projections geometric projections parallelperspective orthographicaxonometricoblique trimetric dimetric isometric cavaliercabinet single-pointtwo-pointthree-point

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Orthographic Transformation Preserves relative dimension The center of projection at infinity The direction of projection is parallel to a principle axis Architectural and engineering drawings

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Axonometric Transformation Orthogonal projection that displays more than one face of an object –Projection plane is not normal to a principal axis, but DOP is perpendicular to the projection plane –Isometric, dimetric, trimetric

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Oblique Parallel Projections Projection plane is normal to a principal axis, but DOP is not perpendicular to the projection plane Only faces of the object parallel to the projection plane are shown true size and shape

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Oblique Parallel Projections www.maptopia.com

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Oblique Parallel Projections

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Typically, is either 30˚ or 45˚ L 1 is the length of the projected side edge –Cavalier projections L 1 is the same as the original length –Cabinet projections L 1 is the half of the original length

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Oblique Parallel Projections Cavalier projections Cabinet projections

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Perspective Projection Pin-hold camera model –Put the optical center (Center Of Projection) at the origin –Put the image plane (Projection Plane) in front of the COP –The camera looks down the negative z axis we need this if we want right-handed-coordinates

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Perspective Projection Projection equations –Compute intersection with PP of ray from (x,y,z) to COP –Derived using similar triangles (on board) –We get the projection by throwing out the last coordinate:

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Homogeneous coordinates Is this a linear transformation?

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Homogeneous coordinates Trick: add one more coordinate: Converting from homogeneous coordinates homogeneous projection coordinates homogeneous viewing coordinates

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Perspective Projection Projection is a matrix multiply using homogeneous coordinates: divide by third coordinate This is known as perspective projection –The matrix is the projection matrix –Can also formulate as a 4x4 divide by fourth coordinate

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Perspective Projection The projection matrix can be much involved, if the COP is different from the origin of the uvn coordinates –See the textbook for the detailed matrix

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Traditional Classification of Projections Three principle axes of the object is assumed –The front, top, and side face of the scene is apparent

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Traditional Classification of Projections

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Perspective-Projection View Volume Viewing frustum –Why do we need near and far clipping plane ?

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Normalizing Transformation Transform an arbitrary perspective-projection view volume into the canonical view volume Step 1: from frustum to parallelepiped

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Normalizing Transformation Transform an arbitrary perspective-projection view volume into the canonical view volume Step 2: from parallelepiped to normalized

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OpenGL 3D Viewing Functions Viewing-transformation function –glMatrixMode(GL_MODELVIEW); –gluLookAt(x0,y0,z0,xref,yref,zref,vx,vy,vz); –Default: gluLookAt(0,0,0, 0,0,-1, 0,1,0); OpenGL orthogonal-projection function –glMatrixMode(GL_PROJECTION); –gluOrtho(xwmin,xwmax, ywmin,ywmax, dnear,dfar); –Default: gluOrtho(-1,1, -1,1, -1,1); –Note that dnear and dfar must be assigned positive values z near =-dnear and z far =-dfar The near clipping plane is the view plane

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OpenGL 3D Viewing Functions OpenGL perspective-projection function –The projection reference point is the viewing-coordinate origin –The near clipping plane is the view plane –Symmetric: gluPerspective(theta,aspect,dnear,dfar) –General: glFrustum(xwmin,xwmax,ywmin,ywmax,dnear,dfar)

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