Introduction to 3d graphics lecture 2 mathematics of the simple camera
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Introduction to 3D Graphics Lecture 2: Mathematics of the Simple Camera. Anthony Steed University College London. Overview. Basic Maths Points Vectors Simple Camera Scenes with spheres COP on +z Local illumination Ambient Diffuse Specular. Overview. Basic Maths Points Vectors

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Introduction to 3d graphics lecture 2 mathematics of the simple camera

Introduction to 3D GraphicsLecture 2: Mathematics of the Simple Camera

Anthony Steed

University College London


Overview

Overview

  • Basic Maths

    • Points

    • Vectors

  • Simple Camera

    • Scenes with spheres

    • COP on +z

  • Local illumination

    • Ambient

    • Diffuse

    • Specular


Overview1

Overview

  • Basic Maths

    • Points

    • Vectors

  • Simple Camera

    • Scenes with spheres

    • COP on +z

  • Local illumination

    • Ambient

    • Diffuse

    • Specular


Basic maths

Basic Maths

  • In computer graphics we need mathematics both for describing our scenes and also for performing operations on it, such as projecting and transforming it.

  • Coordinate systems (right- and left-handed), serves as a reference point.

  • 3 axis labelled x, y, z at right angles.


Co ordinate systems

Co-ordinate Systems

Y

Y

X

X

Z

Z

Left-Handed System

(Z goes in to the screen)

Right-Handed System

(Z comes out of the screen)


Points p x y z

Points, P (x, y, z)

  • Gives us a position in relation to the origin of our coordinate system


Vectors v x y z

Vectors, V (x, y, z)

  • Is a direction in 3D space

  • Points != Vectors

    • Point – Point = Vector

    • Vector+Vector = Vector

    • Point + Vector = Point

    • Point + Point = ?


Vectors v x y z1

y

x

Vector addition

sum v + w

w

v

v + w

Vectors, V (x, y, z)

v

2v

(1/2)V

(-1)v

Scalar multiplication of

vectors (they remain parallel)

w

P

v - w

v

v

w

O

Vector difference

v - w = v + (-w)

Vector OP


Vectors v

Vectors V

  • Length (modulus) of a vector V (x, y, z)

    • |V| =

  • A unit vector


Dot product

Dot Product

  • a · b = |a| |b| cos

    cos = a · b/ |a| |b|

  • a · b = xa ·xb + ya ·yb + za ·zb

  • what happens when the vectors are unit

  • if dot product == 0 or == 1?

  • This is purely a scalar number not a vector


Cross product

Cross Product

  • The result is not a scalar but a vector which is normal to the plane of the other 2

  • direction is found using the determinant

    • i(yvzu -zvyu), -j(xvzu - zvxu), k(xvyu - yvxu)

  • size is a x b = |a||b|sin

  • cross product of vector with it self is null


Parametric equation of a line ray

x(t) = x0 + t(x1 -x0)

y(t) = y0 + t(y1 -y0)

z(t) = z0 + t(z1 -z0)

Parametric equation of a line (ray)

Given two points P0 = (x0, y0, z0) and

P1 = (x1, y1, z1) the line passing through

them can be expressed as:

P(t) = P0 + t(P1 -P0)

=

With -  < t < 


Equation of a sphere

hypotenuse

c

b

a

P

r

yp

xp

(0, 0)

Equation of a sphere

  • Pythagoras Theorem:

  • Given a circle through the origin with radius r, then for any point P on it we have:

a2 + b2 = c2

x2 + y2 = r2


Equation of a sphere1

(x-xc)2 + (y-yc)2 = r2

So for the general case

Equation of a sphere

  • If the circle is not centered on the origin:

We still have

yp

P

(xp,yp)

a2 + b2 = r2

r

b

b

but

yc

a

(xc,yc)

a = xp-xc

b = yp-yc

xp

xc

(0, 0)

a


Equation of a sphere2

Equation of a sphere

  • Pythagoras theorem generalises to 3D giving

Based on that we can easily

a2 + b2 + c2 = d2

prove that the general equation of a sphere is:

(x-xc)2 + (y-yc)2 + (z-zc)2 = r2

x2 + y2 + z2 = r2

and at origin:


Overview2

Overview

  • Basic Maths

    • Points

    • Vectors

  • Simple Camera

    • Scenes with spheres

    • COP on +z

  • Local illumination

    • Ambient

    • Diffuse

    • Specular


Simple camera cross section

Simple Camera (Cross Section)

Y

d

ymax

Z

-Z

COP

ymin


View from the camera

View From the Camera

(xmax, ymax)

(xmin, ymin)


Forming the rays

Forming the Rays

  • Map screen pixels (M by N window) to points in camera view plane

(xmax, ymax)

(M-1, N-1)

(0,0)

(xmin, ymin)


Forming the rays1

Forming the Rays

  • Consider pixel i,j

  • It corresponds to a rectangle

    width = (xmax-xmin)/M

    height = (ymax-ymin)/N

  • Our ray goes through the center of the pixel

  • Thus the ray goes through the point

    (xmin + width*(i+0.5), ymin + height*(j+0.5), 0.0)


Forming the rays2

Forming the Rays

  • Thus the ray from the COP through pixel i,j is defined by

    p(t) = (x(t), y(t), z(t)) =

    (t*(xmin + width*(i+0.5)),

    t*(ymin + height*(j+0.5)),

    t*d-d)


Ray casting

Ray Casting

  • Intersection of Sphere and line (sphere at origin)

  • Substitute the ray equation in the sphere equation and solve!

  • Get an equation in t of the form

    At2 + 2Bt + C = 0


Ray casting1

Ray Casting

If b2 – AC < 0 then the ray doesn’t intersect the sphere.

If b2 -AC = 0 the ray graze (is tangent to the sphere)

If b2 – AC > 0 then there are two roots given by

t = (-b  (b2 – AC))/A

chose the highest value one (the one closest to the COP)


Ray casting2

Ray Casting

  • Intersection of Sphere and line (general case)

    • Sphere is centred at (a,b,c)

    • Translate the start of the ray by (-a,-b,-c)

    • Proceed as before


The image detection

The Image - Detection


Overview3

Overview

  • Basic Maths

    • Points

    • Vectors

  • Simple Camera

    • Scenes with spheres

    • COP on +z

  • Local illumination

    • Ambient

    • Diffuse

    • Specular


Ambient light

Ambient Light

  • Approximation to global illumination

    • Each object is illuminated to a certain extent by “stray” light

    • Constant across a whole object

  • Often used simply to make sure everything is lit, just in case it isn’t struck by light direct from a light source


Ambient light1

Ambient Light

  • Ambient light usually set for whole scene (Ia)

  • Each object reflects only a proportion of that (ka)

  • So far then

Ir = kaIa


Lighting equation 1

Lighting Equation #1

But we use RGB so

Ir, red = ka,redIa,red

Ir,green = ka,greenIa,green

Ir,blue = ka,blueIa,blue


The image ambient

The Image - Ambient


Lambert s law

Lambert’s Law

  • Reflected intensity is proportional to cos 

  • L is the direction to the light

  • N is the surface normal


Diffuse light

Diffuse Light

  • The normalised intensity of the light incident on the surface due to a ray from a light source

  • The light reflected due to Lambert’s law

  • The proportion of light reflected rather than absorbed (kd)


Lighting equation 2

Lighting Equation #2

  • Ambient and diffuse components

  • Again kd is wavelength dependent and we work with kd,red kd,green and kd, blue

Ir = kaIa + kdIi (n.l)


Multiple lights

Multiple Lights?

  • Add the diffuse terms

  • Ii,j is the incoming intensity of light j

  • lj is the vector to light j

m

Ir = kaIa + kdIi,j (n.l j)

j =1


The image diffuse

The Image - Diffuse


Perfect specularity

Perfect Specularity

  • Would almost never see the specular highlight


Imperfect specularity phong

Imperfect Specularity (Phong)

  • E is the direction to the eye

  • N is the normal

  • L is the direction to the light

  • H bisects E and L


Specular component

Specular Component

  • m is the power of the light

    • High m implies smaller specular highlight

    • Low m makes the highlight more blurred

ksIi (h.n)m


Lighting equation 3

Lighting Equation #3

  • Ambient, diffuse&specular components

  • Again if there are multiple lights there is a sum of the specular and diffuse components for each light

    (This is the time to worry about clamping values to 0,1 required for monitor display)

Ir = kaIa + Ii (kd (n.l) + ks(h.n)m )


The image specular

The Image - Specular


Conclusions

Conclusions

  • We can now draw images

    • Forming rays from the camera

    • Intersecting those rays with objects in the scene

    • Colouring the pixels

  • Immediate work required

    • More interesting scenes

    • A useful camera

      • At the moment we must move the objects in front of the camera to be able to see them


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