3d viewing - univie.ac.atvda.univie.ac.at/teaching/graphics/13w/lecturenotes/09_viewing.pdf ·...

© Machiraju/Zhang/Möller

3D Viewing

Introduction to Computer GraphicsTorsten Möller

© Machiraju/Zhang/Möller 2

Reading• Chapter 4 of Angel• Chapter 13 of Hughes, van Dam, …• Chapter 7 of Shirley+Marschner


Objectives• What kind of camera we use? (pinhole)• What projections make sense

– Orthographic– Perspective

• The viewing pipeline• Viewing in WebGL• Shadows

3


3D Viewing• Popular analogy: virtual camera taking

pictures in a virtual world• The process of getting an image onto the

computer screen is like that of taking a snapshot.


3D Viewing (2)• With a camera, one:

– establishes the view – opens the shutter and exposes the film


3D Viewing (3)• With a camera, one:

– establishes the view – opens the shutter and exposes the film

• With a computer, one: – chooses a projection type (not necessarily

perspective) – establishes the view – clips the scene according to the view – projects the scene onto the computer display


The ideal pinhole camera• Single ray of light gets through small pinhole• Film placed on side of box opposite to

pinhole

Projection


The pinhole camera• Angle of view (always fixed)

• Depth of field (DOF): infinite – every point within the field of view is in focus (the image of a point is a point)

• Problem: just a single ray of light & fixed view angle

• Solution: pinhole → lens, DOF no longer infinite


The synthetic camera model• Image formed in front of the camera• Center of projection (COP):

center of the lens (eye)


Types of projections• Choose an appropriate type of projection

(not necessarily perspective) • Establishes the view: direction and

orientation


3D viewing with a computer• Clips scene with respect to a view volume

– Usually finite to avoid extraneous objects• Projects the scene onto a projection plane

– In a similar way as for the synthetic camera model

– Everything in view volume is in focus– Depth-of-field (blurring) effects may be generated


Project onto projection plane

Clip against 3D view volume

Where are we at?





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Projection: 3D → 2D• We study planar geometric projections:

– projecting onto a flat 2D surface – project using straight lines

• Projection rays, called projectors, are sent through each point in the scene from the centre of projection (COP) - our pinhole

• Intersection between projectors and projection plane form the projection


Perspective and parallel projections

• Parallel:– COP at infinity– By direction of

projectors (DOP)

• Perspective:- Determined by COP


COP in homogeneous coordinates

• Perspective projection: COP is a– finite point: (x, y, z, 1)

• Parallel projection– Direction of projection is a vector:

(x, y, z, 1) – (x’, y’, z’, 1) = (a, b, c, 0)– Points at infinity and directions correspond in a

natural way


Perspective vs. parallel• Perspective projection:

– Realistic, mimicking our human visual system– Foreshortening: size of perspective projection of

object varies inversely with the distance of that object from the center of projection

– Distances, angles, parallel lines are not preserved• Parallel projection:

– Less realistic but can be used for measurements– Foreshortening uniform, i.e., not related to distance– Parallel lines are preserved (length preserving?)


Taxonomy of projectionsAngle between projectors and projection plane?

Number of principal axes cut by projection plane


Parallel projections• Used in engineering mostly, e.g., architecture• Allow measurements• Different uniform foreshortenings are

possible, i.e., not related to distance to projection plane

• Parallel lines remain parallel• Angles are preserved only on faces which

are parallel to the projection plane – same with perspective projection


Orthographic (parallel) projections

• Projectors are normal to the projection plane• Commonly front, top (plan) and side

elevations: projection plane perpendicular to z, y, and x axis

• Matrix representation (looking towards negative z along the z axis; projection plane at z = 0):


Orthographic projections• Axonometric projection

– Projection plane not normal to any principal axis– E.g., can see more faces of an axis-aligned

cube


Orthographic projections• Foreshortening: three

scale factors, one each for x, y, and z axis

• Axonometric projectionexample:


Orthographic projection• Axonometric projection – Isometric

– Projection-plane normal makes same angle with each principal axis

– There are just eight normal directions of this kind– All three principal axes are equally

foreshortened, good for getting measurements– Principal axes make same angle in projection

plane• Alternative: dimetric & trimetric (general

case)


Parallel Projection (4)• Axonometric Projection - Isometric

– Angles between the projection of the axes are equal i.e. 120º

• Alternative- dimetric & trimetric


Types of Axonometric Projections

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Oblique (parallel) projections• Projectors are not normal to projection

plane

• Most drawings in the text use oblique projection


Oblique projections• Two angles are of interest:

– Angle α between the projector and projection plane

– The angle φ in the projection plane

• Derive the projection matrix


Derivation of oblique projections


Common oblique projections• Cavalier projections

– Angle α = 45 degrees– Preserves the length of a line segment

perpendicular to the projection plane– Angle φ is typically 30 or 45 degrees

• Cabinet projections– Angle α = 63.7 degrees or arctan(2)– Halves the length of a line segment

perpendicular to the projection plane – more realistic than cavalier


Projections, continued


Perspective projections• Mimics our human visual

system or a camera• Project in front of the center

of projection• Objects of equal size at different distances

from the viewer will be projected at different sizes: nearer objects will appear bigger


Types of perspective projections

• Any set of parallel lines that are not parallel to the projection plane converges to a vanishing point, which corresponds to point at infinity in 3D

• One-, two-, three-point perspective views are based on how many principal axes are cut by projection plane


Vanishing points


• COP at z = 0• Projection plane at z = d

• Transformation is not invertible or affine. Derive the projection matrix.

36

Simple perspective projection


Simple perspective projection• How to get a perspective projection matrix?• Homogeneous coordinates come to the

rescue


Summary of simple projections• COP: origin of the coordinate system• Look into positive z direction• Projection plane perpendicular to z axis





39


General viewing and projections

• With scene specified in world coordinate system

• Position and orientation of camera• Projection: perspective or parallel

– Clip objects against a view volume – which one?

– Normalize the view volume (easier to do this way)

– The rest is orthographic projection


Perspective Projections• Need to describe the viewing

– VRP - view reference point

VRP



– VRP - view reference point– VN - view normal

VRP

VN



– VRP - view reference point– VN - view normal– VUP - up vector

VRPu

v

VUP

VN



– VRP - view reference point– VN - view normal– VUP - up vector– COP - centre of projection

VRPu

v

VUP

VN

COP


Clipping• Projection defines a view volume• add front and back clipping plane • make view volume finite• why is this useful for a perspective

projection?


Clipping (2)• How should we clip a scene against the

view volume? • We could clip against the actual

coordinates, but is there an easier way?


Clipping (3)• Canonical view volume

– Parallel:x = -1, x = 1, y = -1, y = 1, z = 0, z = -1

– Perspective:x = z, x = -z, y = z, y = -z, z = -z_min, z = -1


Viewing Process


Normalization - Parallel• translate the

VRP to the origin


Normalization - Parallel (2)• rotate the view

referencecoordinates suchthat the VPNbecomes the z axis,u becomes the xaxis and vbecomes the yaxis


Normalization - Parallel (3)• shear so that the

direction ofprojection isparallel to thez axis


Normalization - Parallel (4)• translate and

scale into thecanonical viewvolume


Normalization - Parallel (5)• translates the VRP to the origin• rotate the view reference coordinates such

that the VPN becomes the z axis, u becomes the x axis and v becomes the y axis

• shear so that the direction of projection is parallel to the z axis

• translate and scale into the canonical view volume


Normalization - Parallel (6)• Projection matrix:

M = S � T � SH �R� T


Normalization - Perspective• translates the VRP

to the origin


Normalization - Perspective (2)• rotate the view

reference coordinatessuch that the VPNbecomes the zaxis, u becomesthe x axis and vbecomes they axis.


Normalization - Perspective (3)• Translate

so that thecenter of projectionis at the origin


Normalization - Perspective (4)• shear so

that the centreline of the viewvolume becomesthe z axis


Normalization - Perspective (5)• scale into

the canonicalview volume


Normalization - Perspective (6)• translates the VRP to the origin • rotate the view reference coordinates such

that the VPN becomes the z axis, u becomes the x axis and v becomes the y axis.

• translate so that the center of projection is at the origin

• shear so that the centre line of the view volume becomes the z axis

• scale into the canonical view volume


• Projection matrix (perspective):

• Projection matrix (parallel):

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Normalization - Perspective (7)

M = Ppers � S � SH � T �R� T

M = Pparall � S � SH �R� T





62

©Angel/Shreiner/Möller

Computer Viewing• There are three aspects of the viewing

process, all of which are implemented in a pipeline,– Positioning the camera

• Setting the model-view matrix– Selecting a “lens”

• Setting the projection matrix– Clipping

• Setting the view volume

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The WebGL Camera• In WebGL, initially the object and camera

frames are the same– Default model-view matrix is an identity

• The camera is located at origin and points in the negative z direction

• WebGL also specifies a default view volume that is a cube with sides of length 2 centered at the origin– Default projection matrix is an identity

64


Default Projection• Default projection is orthogonal

65

clipped out

z=0

2


Moving the Camera Frame• If we want to visualize an object with both

positive and negative z values we can either– Move the camera in the positive z direction

• Translate the camera frame– Move the objects in the negative z direction

• Translate the world frame• Both of these views are equivalent and are

determined by the model-view matrix– Want a translation Translate(0.0,0.0,-d);– d > 0

66


Moving the Camera• We can move the camera to any desired

position by a sequence of rotations and translations

• Example: side view– Rotate the camera– Move it away from origin– Model-view matrix C = TR

67


The LookAt Function• The GLU library contained the function

gluLookAt to form the required modelview matrix through a simple interface

• Note the need for setting an up direction• Replaced by LookAt() in mat.h

– Can concatenate with modeling transformations• Example: isometric view of cube aligned

with axesmat4 mv = LookAt(vec4 eye, vec4 at, vec4 up);

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gluLookAt• LookAt(eye, at, up)

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Other Viewing APIs• The LookAt function is only one possible

API for positioning the camera• Others include

– View reference point, view plane normal, view up (PHIGS, GKS-3D)

– Yaw, pitch, roll– Elevation, azimuth, twist– Direction angles

70


WebGL Orthogonal ViewingOrtho(left,right,bottom,top,near,far)

near and far measured from camera

71


WebGL PerspectiveFrustum(left,right,bottom,top,near,far)

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Using Field of View• With Frustum it is often difficult to get the

desired viewPerpective(fovy,aspect,near,far) often provides a better interface

aspect = w/h

front plane

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Application of projection: shadows

• Essential component for realistic rendering• Naturally use projections

– Can produce hard shadows– Only handles shadows on a plane– To shadow on a polygonal face, need clipping

• More advanced shadow algorithms exist, e.g., soft shadows and penumbra (not easy to do)

• Blinn 74


Shadow via projection• A shadow polygon is obtained through

projection where the center of projection is a light source


Shadow polygon: parallel projection

(xp, yp, zp)

(xs, ys, 0)

• Project shadow on z = 0• Light at ∝ (directional light)• Derive the projection matrix


• Light at infinity:

• hence projection:

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Shadows: parallel projection

Now draw the objecton the ground plane


• Project on plane z = 0• Point light source • Derive the matrix

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Shadows: perspective projection


• Local Light:

• projection matrix:

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Shadows: perspective projection


Shadow of a teapot


Shadows• requires no extra memory• easily handles any number of light sources• only shadows onto ground plane• cannot handle objects which shadow other

complex objects• every polygon is rendered N+1 times, where

N is number of light sources

3d viewing - univie.ac.atvda.univie.ac.at/teaching/graphics/13w/lecturenotes/09_viewing.pdf ·...

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