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Geometry: �Projection

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Instructor: Christopher Rasmussen (cer@cis.udel.edu)

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February 25, 2021 ❖ Lecture 4

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Outline

  • Projections, special views
    • Parallel/Orthographic
    • Perspective
  • GLM
  • HW #1
  • Marschner readings: 7.3, 7.5

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Projections in CG

  • Mapping from n-D space down to lower-dimensional subspace
    • E.g., point in 3-D space to point on plane (a 2-D entity) in that space
    • We will be interested in such 3-D to 2-D projections where the plane is the image plane
  • Things to know:
    • Where is the plane? Derived from view matrix V
    • What region of the plane does the image occupy?
    • What kind of projection?

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Parallel Projections

3-D points "slide" along direction d until they hit plane, where their locations can be described in 2-D. For example, Q→ q, P→p *

from Hill

*not our usual notation for points

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Parallel Projection subtypes

Oblique: d in general orientation

relative to image plane normal n

Orthographic: d parallel to n

from Hill

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Oblique Projection: Example

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Orthographic Projection

  • Projection direction d is aligned with Z axis
  • Viewing volume is “brick”-shaped region in space
    • Not the same as image size
  • No perspective effects—distant objects look same as near ones, so camera (x, y, z) ⇒ image (x, y)

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from Hill

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Camera Orientation under Orthographic Projection (set by lookat)

  • Plan: Projection onto XZ (ground) plane (aka: ichnographic view)
  • Elevations
    • Front: Projection onto XY plane
    • Side: Projection onto YZ plane

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courtesy of http://glasnost.itcarlow.ie/~powerk/GeneralGraphicsNotes/projection/orthographicprojection.html

Z

X

Y

Da Vinci's 1502 plan of Imola

Elevations, plan of Pantheon in Rome

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Plan/elevation views in games (not necessarily orthographic)

Z

X

Y

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Camera Orientation under Orthographic Projection (set by lookat)

  • Isometric: "¾ view" -- axes equally scaled
  • Dimetric: 2 axes the same, 1 different
  • Trimetric: All axes different

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By SharkD - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=8497328

These angles are not the same as Euler angles (roll, pitch, yaw) of camera. Camera angles for isometric view are explained here

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Camera Orientation under Orthographic Projection (set by lookat)

  • Isometric: "¾ view" -- axes equally scaled
  • Dimetric: 2 axes the same, 1 different
  • Trimetric: All axes different (aka general view)

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some images courtesy of http://glasnost.itcarlow.ie/~powerk/GeneralGraphicsNotes/projection/orthographicprojection.html

SimCity 2000 (not quite isometric)

Diablo

Zaxxon

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The Geometry Pipeline so far...

M = rotation, scaling, translation that puts model where we want it in the world

V = transformation that places the camera (the result of the lookat() function)

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Model vs. World Coordinates

M = rotation, scaling, translation that puts model where we want it in the world

Z

X

Y

Z

X

Y

model frame

world frame

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Model vs. World Coordinates

Mi for each pencil i to scale, rotate,

and translate it just so

Z

X

Y

world frame

image courtesy Osman Usta, S2019 student

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Orthographic Projection in OpenGL

  • Setting up the viewing volume (VV) with GLM library
    • glm::ortho()
      • left, right, bottom, top: Coordinates of sides of viewing volume (in camera coordinates)
      • znear, zfar: Distances to front, back sides of VV
        • Negative = Behind camera
    • glm::ortho2D(): glm::ortho() with near = -1, far = 1

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Projection in OpenGL

  • Applied after model M and view V transformations have put things in camera coordinates
  • Actual matrix scales VV to canonical VV (CVV): Cube extending from -1 to 1 along each dimension
    • This is properly a transformation; the projection is accomplished later by stripping off the Z coordinate
  • All together, called "MVP" transform -- as matrices, P * V * M * x

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P = intrinsics of camera; result sometimes called "clip coordinates"

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Orthographic Projection Matrix for CVV

Translation to origin followed by scaling to 2 x 2 x 2 cube

(If n & f are not distances then n > f, which is why it is

“-2/(f - n)” below — see formula in Marschner):

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Orthographic Projection Matrix for CVV

Translation to origin followed by scaling to 2 x 2 x 2 cube

(If n & f are not distances then n > f, which is why it is

“-2/(f - n)” below — see formula in Marschner):

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Perspective with a Pinhole Camera (i.e., no lens), aka "camera obscura"

from Forsyth & Ponce

Instead of single direction d characteristic of parallel projections,

rays emanating from single point c (or eye e) define

perspective projection

c

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Stenop.es project: Apartment as pinhole camera

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More Stenop.es (note projection is upside-down)

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Perspective Projection

from Forsyth & Ponce

c

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Contrasting VVs for perspective and orthographic projection

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Issues with Real Pinhole Cameras

  • It takes a long time to gather enough light through a small pinhole -- not suitable for scenes with motion
    • Bigger pinhole (aka aperture) for more light → Blurry image
    • Smaller pinhole for greater resolution → Diffraction limit
  • A suitable lens can admit more light while reducing blur

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from https://en.wikipedia.org/wiki/Camera_lens

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Perspective Projection: Viewing Volume

  • Characteristic shape is a frustum—a truncated pyramid
    • Far plane is arbitrary

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Perspective Projection: From Camera Coordinates to Image Coordinates

xcam

xim

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Perspective Projection

  • Letting the camera coordinates of the projected point be xcam = (x, y, z)T leads by similar triangles to:

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Focal length

(“near” distance)

y / z = v / f and x / z = u / f, so....

v = fy / z and u = fx / z

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Perspective Projection

  • Letting the camera coordinates of the projected point be xcam = (x, y, z)T leads by similar triangles to:

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Focal length

(“near” distance)

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Perspective Projection Matrix

  • Using homogeneous coordinates, we can describe a perspective transformation with the image plane at z = -f (because f > 0 but z < 0) via a 4 x 4 matrix multiplication:

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Last step accomplishes distance-dependent scaling by the rule for converting between homogeneous and regular coordinates. This is called the perspective division

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Perspective Transformation Matrix

  • Intuitively, what we’re doing is transforming the viewing frustum into a “brick” (aka axis-aligned box), then doing orthographic projection

f does not mean

same thing here

as on previous slide!!

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Perspective Transformation: Details

  • Instead of simply projecting all z to near plane n, set matrix such that z = n maps to n and z = f maps to f (this is perspective matrix in Marschner):

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  • After applying this, multiply by orthographic transform to scale everything to CVV and project to image coordinates
    • Project only after any steps that require depth information

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Perspective Projections in OpenGL

  • glm::frustum() sets transformation to CVV
    • Arguments like glm::ortho() (set left, right, bottom, top, near, far directly) but near, far are distances and must be positive
    • Vertical field of view (FOV): θ = 2 arctan(0.5 height / near), where height = top - bottom

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from Woo et al.

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glm::perspective()

  • Simplifies call to glm::frustum()
  • Arguments:
    • fovy: Field of view angle (degrees) in Y direction
    • aspect: Ratio of width to height of viewing frustum
    • near, far: Same as glm::frustum()

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from Woo et al.

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Field of View

  • Controls “strength” of perspective effects

courtesy of S. Marschner

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Field of View

  • Controls “strength” of perspective effects

courtesy of S. Marschner

Closer to orthographic projection because rays are more nearly parallel

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Viewport Transform

  • A final transform shifts normalized device coordinates (NDC—what you have after perspective divide) to image coordinate origin and scales to fit window width & height
    • This is what makes x = 0, y = 0 go to corner of window instead of center
    • More later in the semester...

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Geometry pipeline

Coordinate change rigid transform / view transform V

Perspective transform P

(and/or orthographic scaling) to CVV

2-D scale and shift

Perspective division

Orthographic projection

Camera coordinates

Clip coordinates

Normalized device coordinates

Window coordinates

Screen coordinates

World coordinates

Rotate, scale, translate with model transform M

Model coordinates

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Combining Transformations: Order

  • Model-View-Projection matrices combined to make single "MVP" matrix:
    • Model -> World coordinates
    • World -> Camera coordinates
    • Camera -> Screen coordinates
  • But...flip order!
    • P*V*M*x

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Transformations with GLM

  • Manual with full API (i.e., how to call functions) is here: http://glm.g-truc.net/glm.pdf

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#include <glm/glm.hpp>

#include <glm/gtx/transform.hpp>� � glm::mat4 myMatrix;� glm::vec4 myVector;

// fill these somehow� glm::vec4 transformedVector = myMatrix * myVector;

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More next time on OpenGL in general

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Important Transformations with GLM

  • Model transformations
    • glm::translate()
    • glm::scale()
    • glm::rotate()

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  • View transformations
    • glm::lookat(eye, center, up)

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  • Projection transformations
    • glm::perspective() // like gluPerspective()
    • glm::ortho() // like glOrtho()

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Link to API

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Model Transformations with GLM

  • glm::translate(t)
  • glm::scale(s)
  • glm::rotate(angle, axis)

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  • glm::mat4 Model =

glm::translate(glm::vec3( 3.0f,

0.0f,

0.0f));

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Model Transformations with GLM

  • glm::translate(t)
  • glm::scale(s)
  • glm::rotate(angle, axis)

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  • glm::mat4 Model =

glm::rotate((float) M_PI,

glm::vec3( 0.0f,

0.0f,

1.0f));

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View Transformation with GLM

  • glm::lookat(eye, center, up)

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  • glm::mat4 View =

glm::lookat( glm::vec3(4,3,3),

glm::vec3(0,0,0),

glm::vec3(0,1,0));

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Link to API

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Projection Transformations with GLM

  • glm::perspective(fovy, aspect, near, far)
  • glm::ortho(left, right, bottom, top, zNear, zFar)

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  • glm::mat4 Projection =

glm::perspective( 45.0f,

4.0f / 3.0f,

0.1f,

100.0f);

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Homework #1

  • Part 1:
    • Get OpenGL 3.3, GLFW installed using instructions in OpenGL Tutorial #1 regarding installation on your system
    • Run the program, which just pops up an empty (black) window

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Homework #1

  • Part 1:
    • Get OpenGL 3.3, GLFW installed using instructions in OpenGL Tutorial #1 regarding installation on your system
    • Run the program, which just pops up an empty (black) window
  • Part 2: Extend modified version of OpenGL Tutorial #3 (Matrices) as follows:
    • Create new 3-D polyhedron from transformed versions of triangle and/or square
    • Make 2-D pattern of different simple shapes/objects on XZ plane
      • Use a few different colors
    • Create plan view and approximately isometric view of model with different parameters to lookat(), submit screenshots with orthographic projection

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Homework #1

  • Part 1:
    • Get OpenGL 3.3, GLFW installed using instructions in OpenGL Tutorial #1 regarding installation on your system
    • Run the program, which just pops up an empty (black) window
  • Part 2: Extend modified version of OpenGL Tutorial #3 (Matrices) as follows:
    • Create new 3-D polyhedron from transformed versions of triangle and/or square
    • Make 2-D pattern of different simple shapes/objects on XZ plane
      • Use a few different colors
    • Create plan view and isometric view of model with different parameters to lookat(), submit screenshots with orthographic projection
  • Grad section: Programmatic generation of more complex pattern

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Selected HW #1 submissions (spring, 2019)

Scott Benton (440)

Osman Usta (640)