컴퓨터 그래픽스를 위한 기하학적 기초 (Line Geometry for Computer Graphics)

Slides:

Advertisements

Similar presentations

Vector Equations in Space Accelerated Math 3. Vector r is the position vector to a variable point P(x,y,z) on the line. Point P o =(5,11,13) is a fixed.

Advertisements

An Optimized Soft Shadow Volume Algorithm with Real-Time Performance Ulf Assarsson 1, Michael Dougherty 2, Michael Mounier 2, and Tomas Akenine-Möller.

Computer Graphics1 Geometry Area of polygons & Volume Of Polygonal surfaces.

Recovery of affine and metric properties from images in 2D Projective space Ko Dae-Won.

Computer Graphics Lecture 8 Arbitrary Viewing II: More Projection, Clipping and Mathematics of 3D Viewing.

1 Computer Graphics Chapter 8 3D Transformations.

1 Clipping. 2 Transformation Sequence X Y Z X Y Z X Y Z X Y Z Object Coords. Eye Coords. Clip Coords. Normalized Device Coords. Screen Coords. Implementation:

MA Day 9 – January 17, 2013 Review: Equations of lines, Section 9.5 Section 9.5 –Planes.

Projective geometry Slides from Steve Seitz and Daniel DeMenthon

Lecture 15: Single-view modeling CS6670: Computer Vision Noah Snavely.

1 Chapter 5 Viewing. 2 Perspective Projection 3 Parallel Projection.

1 of 23 New Lecture And Lab Information Lectures: –Thursday 12:00 – 13:00 (room tba) –Friday 15:00 – 16:00 (A28) Labs: –Wednesday 10:00 – 11:00 (A305)

Project Geometry Jiecai He (Jake)

Section 9.4: The Cross Product Practice HW from Stewart Textbook (not to hand in) p. 664 # 1, 7-17.

1 1.9 © 2016 Pearson Education, Inc. Linear Equations in Linear Algebra THE MATRIX OF A LINEAR TRANSFORMATION.

COMP 175: Computer Graphics March 24, 2015

Copyright  1999 by James H. Money. All rights reserved. Except as permitted under United States Copyright Act of 1976, no part of this publication may.

CS559: Computer Graphics Lecture 9: Projection Li Zhang Spring 2008.

Computer Vision Lecture #2 Hossam Abdelmunim 1 & Aly A. Farag 2 1 Computer & Systems Engineering Department, Ain Shams University, Cairo, Egypt 2 Electerical.

Metrology 1.Perspective distortion. 2.Depth is lost.

OpenGL Conclusions OpenGL Programming and Reference Guides, other sources CSCI 6360/4360.

CAP4730: Computational Structures in Computer Graphics 3D Transformations.

Transformational Geometry CS418 Computer Graphics John C. Hart.

Viewing CS418 Computer Graphics John C. Hart. Graphics Pipeline Homogeneous Divide Model Coords Model Xform World Coords Viewing Xform Still Clip Coords.

Geometric Algebra 4. Algebraic Foundations and 4D Dr Chris Doran

Computer Graphics Camera Projection / Picking CO2409 Week 8 - Optional Advanced Material Not on Exam.

click to start Example: A LINEAR TRANSFORMATION.

Rendering Pipeline Fall, D Polygon Rendering Many applications use rendering of 3D polygons with direct illumination.

Geometric Transformations UBI 516 Advanced Computer Graphics Aydın Öztürk

3-D Transformational Geometry CS418 Computer Graphics John C. Hart.

CS552: Computer Graphics Lecture 11: Orthographic Projection.

컴퓨터 그래픽스를 위한 기하학적 기초 (Line Geometry for Computer Graphics)

Computational Geometry 2012/10/23. Computational Geometry A branch of computer science that studies algorithms for solving geometric problems Applications:

CS559: Computer Graphics Lecture 9: 3D Transformation and Projection Li Zhang Spring 2010 Most slides borrowed from Yungyu ChuangYungyu Chuang.

Teacher Orientation Linear Equations in two Variables Introduction Conditions with two lines in a plane Methods to solve LEs.

1.1 The row picture of a linear system with 3 variables.

Computer Graphic Creator: Mohsen Asghari Session 4 Fall 2014.

Transformations Szirmay-Kalos László. Transformations Can destroy our equations Limit the transformations and the shapes and require invariance – Line.

CS552: Computer Graphics Lecture 9: Perspective Projection.

CS552: Computer Graphics Lecture 12: 3D Clipping.

Lecture 10 Geometric Transformations In 3D(Three- Dimensional)

Lec 23: Single View Modeling

3. Transformation

Algebra Review Systems of Equations page 69

8.7Systems of Linear Equations – Part 1

Prof. Lizhuang Ma Shanghai Jiao Tong University

Lesson 3-6: Perpendicular & Distance

CSCE 441 Computer Graphics 3-D Viewing

Equations of Lines.

Lecture 7 Geometric Transformations (Continued)

Intersection between - Lines, - Planes and - a plane & a Line

Lines and Planes in Space

Viewing Transformations

Systems of Equations Solving by Graphing.

Computer Graphics Lecture 20

Clip against view volume Project onto projection plane

3.1 Notes: Solving Systems of Equations

Systems of Equations Solving by Graphing.

Points, Lines, and Planes QUICK DRAW FOR POINTS!

Prof. Lizhuang Ma Shanghai Jiao Tong University

Systems of Equations Solving by Graphing.

THREE-DIMENSIONAL VIEWING

컴퓨터 그래픽스를 위한 기하학적 기초 (Line Geometry for Computer Graphics)

1.2 Solving Linear Systems by Graphing

Vector Equations in Space

Chapter 9 Lesson 3 Pg. 699 Solving Systems of Equations by Graphing

3.6 Parallel Lines in the Coordinate Plane

Chapter 9 Lesson 3 Pg. 699 Solving Systems of Equations by Graphing

Presentation transcript:

컴퓨터 그래픽스를 위한 기하학적 기초 (Line Geometry for Computer Graphics) 서울대학교 컴퓨터공학부 김명수

View Volume

3D Culling and Clipping

Viewing Transformation

Do you remember?

Line Equation in 2D

Line in Plane

Line in Homogeneous Coords

Line from Two Points in Plane

Using Homogeneous Coord

Example

Intersecting Two Lines

Example

Plane in Space

Plane by Three Points

Plane by Three Points

Wedge Product

Point by Three Planes

Perspective Projection in 2D

Its Matrix Form

Parallel Projection in 2D

Its Matrix Form

Projection in 3D

Projection in 3D

Projection in 3D

Projection in 3D

Perspective Projection in 3D

Parallel Projection in 3D

Transformation to X-axis

Its Geometric Meaning

Transformation to XY-Plane

Conclusions (Line Geometry in Computer Graphics) Duality Between Points and Lines in 2D Duality Between Points and Planes in 3D Intersecting Two Lines (or Three Planes) using Cross Products (or Wedge Products) Simple Viewing Transformation Matrix Perspective and Parallel Projection in the Same Way