lecture 15: perspective projection pipeline

Lecture 15:Perspective Projection Pipeline

October 27, 2020

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge Slide 2

How do Pixels get Colored?§ Step back and consider two alternatives

for (each_object) render (object)

for (each_pixel) color (pixel)

o Reasoning goes “What pixels are altered by this object?”

o Basis for OpenGL rendering pipeline.

o Highly Parallel on modern hardware.

o Fast, but object interactions extremely hard.

o Reasoning goes “What object alter this pixels coloring?”

o Basis for Ray Casting and Ray Tracing.

o Highly Parallel

o Cost mainly intersecting 3D objects with light rays cast back into the scene.

Fill Pixels Inside Triangles

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 3

Fifty years ago “Scan Conversion”

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 4

A Characterization of Ten Hidden-Surface AlgorithmsIvan E. Sutherland, Robert F. Sproull, and Robert A. SchumackerACM Computer. Surveys. 6, 1 (March 1974)

Forty seven years ago, draw triangles so closest occludes

all others.

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Thirty seven years ago - foundation of Silicon Graphics and beginning of OpenGL.

In Essence: The Perspective Projection

Pipeline

Four Major Tasks• Think of rending objects in terms of:–Modeling– Geometry Processing– Rasterization– Fragment Processing

Modeling GeometricProcessing Rasterization Fragment

Processing

Frame BufferToday’s Lecture

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Moving to Formulation #3• Review, #1 origin at PRP (Eye).• Review, #2 origin at image center.• Review, #1 and #2– No useful information on the z-axis

• We now have a new goal• Project into a canonical view volume• A rectangular volume with bounds:– U: -1 to 1, V: -1 to 1, D: -1 to 1

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 7

Remember When We Started• What happens if you multiply a point in

homogeneous coordinates by a scalar?• Nothing!

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 8

xyzw

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Scalar Multiplication Continued• Multiply a homogeneous matrix by a scalar?• Again, nothing changes.

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 9

ax + by+ cz+ dwex + fy+ gz+ hwix + jy+ kz+ lwmx + ny+oz+ pw

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ax + by+ cz+ dwex + fy+ gz+ hwix + jy+ kz+ lwmx + ny+oz+ pw

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s ax + by+ cz+ dw( )s ex + fy+ gz+ hw( )s ix + jy+ kz+ lw( )s mx + ny+oz+ pw( )

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Form #1 & Textbook Derivation

10/22/20CSU CS 410, Fall 2020, © Ross Beveridge 10

Lecture Form #1

1 0 0 00 1 0 000

00

1#1 𝑑

00

𝑑 0 0 00 𝑑 0 000

00

𝑑1

00

Equivalent

Now introduce clipping planes• Text introduces n, the near clipping plane.• Also introduces f, the far clipping plane.• Sets what first called d to n. • The handling of z now carries information?

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 11

n 0 0 00 n 0 00 0 n+ f − fn0 0 1 0

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#

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nxny

zn+ zf − fnz

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n 0 0 00 n 0 00 0 n+ f − fn0 0 1 0

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xyz1

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Visualize View Volume (View 1)

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Visualize View Volume (View 2)

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 13

Consider Some Key Points

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00𝑛

00𝑓

11𝑓1

1𝑛

Formulation #3: The Algebra

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The 1,1 corner of the near clipping plane maps to 1,1 on image plane.The 1,1 corner on the far clipping plane comes toward the optical axis and becomes n/f on the image plane.

𝑀!

Visualize Frustum Change

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 16

SageMath Notebook CS410lec15n01

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Geometry Pipeline

Understand each of these!

SageMath Notebook CS410lec17n01

10/22/20 CSU CS 410, Fall 2020, © Ross Beveridge 18

CanonicalView

Volume

Image Plane Looking Down