Information about 3 d viewing

Three-dimensional Viewing Pipeline Transform into view coordinates and Canonical view volume Clip against canonical view volume Project on to view plane Map into viewport Transform to physical Device coordinates transform clip transform World coordinates(3D) View coordinates(3D) View coordinates(3D) View coordinates(2D) Normalized device coordinates Physical device coordinates

Parallel Projection Mostly used by drafters and engineers to create working drawings of an object which preserves its scale and shape. The distance between the COP and the projection plane is infinite i.e. The projectors are parallel to each other and have a fixed direction. P1 P2 P1‟ P2‟ Projection plane

• Orthographic projection: here direction of projection is perpendicular to the view plane. • Axonometric projection: when direction of projection is not parallel to any of three principal axes.

Perspective Projection • Generalization of the principles used by artists in drawing of scenes. • It takes object representation in view space (L.H.S.) and produce a projection on the view plane (canvas used by the artist). • The projection of a 3D point onto the viewplane is the intersection of the line from the point to the COP (eye position of the artist).

Cont. • Distance between the COP and projection plane is finite. • Perspective projection does not preserve object scale and shape. P1 P2 P1‟ P2‟ Projection plane COP

Perspective Anomalies Perspective foreshortening - the farther an object from the COP, the smaller it appears (i.e. its projected size becomes smaller). cop View plane

Vanishing points - there is an illusion that certain sets of parallel lines (that are not parallel to the view plane ) appear to meet at some point on the view plane. - the vanishing point for any set of parallel lines that are parallel to one of principal axis is referred to as a principal vanishing point (PVP). - the number of PVPs is determined by the number of principal axes intersected by the view plane.

One principal vanishing point projection - occurs when the projection plane is perpendicular to one of the principal axes (x, y or z ). Vanishing point View plane is parallel to XY-plane and intersects Z-axis only.

Two principal vanishing point projection X-axis Vanishing point Z-axis Vanishing point View plane intersects Both X and Z axis but not the Y axis

• Three principal vanishing point intersection View plane intersects all Three of the principal axis X, Y and Z axis VP1 VP3 VP2

Deriving Perspective Projection Assume point vertex denoting COP : (xc,yc,zc) point on the object : (x1,y1,z1) representation of “projection ray” containing above two points x = xc + ( x1-xc) u …………..eq 1 y = yc + ( y1-yc) u …………..eq 2 z = zc + ( z1-zc) u …………..eq 3 The projected point (x2, y2,D) will be the point where this line intersects the XY plane . Putting z=0 for this intersection point in eq 3 . u = - zc / z1-zc

Substituting into first two equations, x2 = y2 = Value of D may be computed which is different from zero (to preserve depth relationship between objects) D = z1 / (z1 –zc) z-z zx-zx c1 c11c z-z zy-zy c1 c11c

Standard perspective projection y x z A (x,0,z) A‟ (x‟,0,0) P (x ,y, z) P‟ (x‟,y‟,0) C(0,0,-d) do (0,0,0)B (0,0,z) z

Using similar triangles ABC and A‟OC, x‟ = d.x / (z+d) y‟ = d.y / (z+d) z‟ = 0 matrix representation : d100 0000 00d0 000d

viewing based on synthetic camera analogy. Specifying an arbitrary 3D view

By selecting different viewing parameters, user can position the synthetic camera. View reference point View-up vector View plane

Effect of change of viewing parameters Imagine a string tied to „view reference point‟ on one end and to the synthetic camera on the other end. By changing viewing parameters, we can swing the camera through the arc or change the length of the string. - changing the view distance is equivalent to how far away from the object the camera is when it takes the picture. - changing the view reference point will change the part of the object that is shown at the origin.

Cont. - changing the view plane normal is equivalent to taking photograph of object from different orientations. - changing view-up is equivalent to twisting the camera in our hands. It fixes the camera angle.

View Volume - The view volume bounds that portion of the 3D space that is to be clipped out and projected onto the view plane.

View Volume for Perspective Projection - its shape is semi-infinite pyramid with apex at the view point and edge passing through the corners of the window. cop View window Front clipping plane Back clipping plane Frustum view volume

View Volume for Parallel Projection -It's shape is an infinite parallelepiped with sides parallel to the direction of projection. Parallelepiped Viewed volume View window Front clipping plane Back clipping plane

Producing a Canonical view volume for a perspective projection cop View window Front clip Back clip View frustum centerlineGeneral shape for the Perspective View volume View volume

Step 1: shear the view volume so that centerline of the frustum is perpendicular to the view plane and passes through the center of the view window. Frustum centerline View volume

Step2: scale view volume inversely proportional to the distance from the view window, so that shape of view volume becomes rectangular parallelepiped. View volume

Converting object coordinates to view plane coordinates similar to the process of rotation about an arbitrary axis zw Yw xw World coordinate system Yv Xv VRP View plane (eye) coordinate system

Steps: 1. Translate origin to view reference point (VRP). 2. Translate along the view plane normal by view distance. 3. Align object coordinate‟s z-axis with view plane coordinates z- axis (the view plane normal). a)- Rotate about x-axis to place the line (ie. Object coordinates z-axis) in the view plane coordinates xz-plane. b)- Rotate about y-axis to move the z axis to its proper position. c)- Rotate about the z-axis until x and y axis are in place in the view plane coordinates.

Ref: „Computer Graphics‟ by S. Harrington (pp. 279-284)

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