09 july 2013 update

1 Pelican Aero Group

09 July 2013 update

Trigons, Normals, and QuadragonsVector-based Solutions to the Classic

Problems of Computer Graphics

Notes / Options:Slideshow animated (F5)See also slide footnotes

J. Philip Barnes

2 Pelican Aero Group Trigons, Normals, and Quadragons J. Philip Barnes 09 July 2013 www.HowFliesTheAlbatross.com

Overview

Presentation Contents • “Flexible-cylinder” geometry grid • Geometry math modeling tools• Vectors ~ history & key operations • Fast world-to-pyramid transformation• Review & Renew: Rotation matrices • Review & Renew: Z-buffer & Ray-casting • Vector solution : “Point-in-polygon” • Vector solution: Inter-vertex interpolation• Vector solution: Constant-curvature Tesselation• “Pyranometer” ambient lighting model• Apply ~ "Render Raptor" & “TrigonoSoar”

Display a 3D-math-modeled shape on a 2D screen


“Flexible Cylinder” Geometry Grid

Computer graphics classic problem 1:Generate and manage 3D geometry

Arrays and SubscriptsObject (n), Row (i), Col (j),Vertex (k), Quadragon (m)

l

qlq

z

x

y

Quadragon

VertexNormal

Vertex


Geometry Math-model Building Blocks

Pseudo-Cosine

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

e y / ymax

= cos m (hp/2)

h x / xmax

m = 2.0 1.0 0.5

Exponential

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

e = e-5h m

Pseudo-Sine

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

e = sin (hm p)

h

Varabola

0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0

e = h m

h


Review, Renew, Theory, & History* of the Casteljau-Bézier Curve • P. De Casteljau first conceived (1963) today’s “Bézier Curve”

• His notes & algorithm remained proprietary at Citroën for years• Set (Np) Control points for control polygon near, not on, curve• (Np-1, Np-2,...) interpolation series on polygon-sides for x(t), y(t)

• De Casteljau’s Algorithm is represented by the Bernstein Basis• as discovered by A.R. Forrest ~ Geometric construct → math

• S. Bernstein’s polynomial (B) derivative (dB/dt) is introduced herein• P. Bézier independently conceived (1966) the control polygon

• Weight factors not widely used (differ from Casteljau/Bernstein) • Bézier’s work at Rénault is the foundation of UNISURF & CATIA

* R.T. Farouki, The Bernstein polynomial basis: a centennial retrospective, UC Davis, 03 Mar 2012 * G. Farin, A History of Curves and Surfaces in CAGD, Arizona State University, 2007

t = 1/3 p1

p2

pNp

p3

x

y 1/3

1/31/3 x,y

Animated graphic (shift F5) [dB/dt][dc/dt](t)

notes) chart (see Bc(t)

:parametric are (dc/dt) rate & w...z,y,x,(c) coordinate any

(p) "points, control" 2N For

pN

1ii

pN

1ii

p

pi

pi

c

c

11

2

11

1

11

1

11

ipNip

ipNi

ii

pNiii

pi

ttiN

tti

adtdBttaB

iNiN

a p

)()(

)()(

)/()(

)!()!()!(


0.0 0.2 0.4 0.6 0.8 1.00.0

0.2

0.4

0.6

0.8

1.0Np = 6 points, Yp(Xp)

Xp

Yp

2

34

5 6 1

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0 Bernstein PolynomialsB1B2B3B4B5B6

t

Bi

Area = 1/Np

SB i = 1

t

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00.0

0.2

0.4

0.6

0.8

1.0Resulting x(t), y(t)

t

x = S Bi Xpi

y = S Bi Ypi

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0-2.0

-1.0

0.0

1.0

2.0

3.0 "Velocities" dx/dt, dy/dt

t

dx/dt = S (dB/dt)i Xpi

dy/dt = S (dB/dt)i Ypi

Demo: Casteljau-Bézier 2D Curve Example

V = (dx/dt) i + (dy/dt) j

N=VxkV


“TrigonoSoar” Math-modeled 3D Surface

Chord, thickness, twist, etc. parametric along the spar from centerline to winglet tip

xyz

Aero axes


Math-modeled Aircraft

AlgebratrossWandering Albatross

RegenoSoarRegenerative-soarer


“RenderRaptor” ~ Wireframe & Flat-Shade


z

x

yB

A

g

C

c

ik

j

Unit vectorsa A / |A|b B / |B|c C / |C|

cos g = a bsin g = |axb|

Vectors ~ Origin and Key Operations

J. Willard Gibbs Oliver Heaviside


zx

y

B

A

C=AxB

ik

j

Vector Cross Product ~ Elegant Visual Derivation

Add cross products, each axisImportant ~ Retain order A:B

Cx = Ay Bz - Az By Cy = Az Bx - Ax Bz Cz = Homework !

z, k x,i

y, j

Ax

Bx

ByAy

Az

Bz

To obtain the cross product AxB above, simply add the orthogonal

cross product components as shown at upper right


n

e f

e

e f cosn

Trigons, Normals, and Quadragons J. Philip Barnes 09 July 2013 www.HowFliesTheAlbatross.com

Fast World-to-Pyramid Transform & 90o Exclusion Criteria

f

e

y-ye

k

z-ze

1. cosn = |zf - ze| / ef ; sinn = -(yf - ye) / ef 2. sine = (xf - xe) / (ef cosn) ; cose = (zf - ze) / (ef cosn) 3. Rotate (e) about y-ye axis through point (e)4. Rotate (n) about x-xe axis through point (e)5. Optionally rotate (k) for “cockpit-roll” effect6. Project to screen (perspective transformation)

v

xs, zs

xp

yp

nx-xe

Graphics classic problem 2:Efficiently project any vertex (v) in 3D onto 2D screen, given: eye point (e) and focus point (f) Approach:Twice rotate the world about axes through (e), operating on coordinate differences (x-xe etc.)

First test a new "90-deg exclusion" criterion. Skip transformations for any quadragon which, after transformation, would have all vertices (v) at least 90o from the line of sight, based on the angle included by the vectors ev & ef using only the dot-product sign. This adds 3 multiplications (xv-xe)(xf -xe) + (yv-ye)(yf -ye) + (zv-ze)(zf -ze) for all vertex coordinates, but avoids at least 9 to transform excluded vertices & unit-normals.

FocuspointEye

point ef

After world

rotations

evBefore world

rotations

View Pyramid

e

Chart origin: JPB, 2007

Animated graphic (shift F5)


Rotation Matrix Concatenation ~ Review and Renew

) j(i, between"" )(k subscript 3 Note

:matrices 3x3 two of ionConcatenat) , ,( rotations pyramidtoWorld

) , ,(yaw & pitch, roll, Manuever

rd

3

1

;k

kjikij BACBAC

knensApplicatio&Reviewproduct -Matrix

worlde

e

e

wp

pyramide

e

e

k LLjkLikijwp

zzyyxx

Tzzyyxx

whereT

enk

kne

:)(

:][T transform pyramid-to-world the Apply];[ matrix rotation roll" cockpit" optional an &

][ ],[ matrices rotation north & east Defineleft. to right proceed ionsconcatenat Rotation

wp

3

1

3

1

transform pyramid-to-world:nApplicatio

3

1

3

1

3

1

3

1

3

1

k LLjkLik

k LLjkLikij

kkjikij

CBACBAD

BCAD

BCACBAD

)(

:left-to-right matrices, 3 eConcatenat

3

1

3

1

3

1

3

1

3

1

3

1

3

1

)(

)()(

kmjLm

L mkLikij

mjk L m

LmkLikij

kkjikij

DCBAE

DCBAE

BCDAE

BCDADCBAE

:therefore and

:left-to-right matrices, 4 eConcatenat

kkkk

k

k

nnnnn

n

ee

eee

e

1 0 00 cos sin0 sin- cos

][

:(e) through axis-z , )( roll" cockpit" :Optioncos sin 0sin- cos 00 0 1

][

:(e) through axis-x an about )( rotate Then,cos 0 sin-0 1 0

sin 0 cos][

:(e) through axis-y a about ) ( rotate First,



Demo: object-maneuver and world-pyramid rotations

Stationary object, moving observer

Fixed observer, maneuvering object


Screen Plane

Polygon Pipeline Overview (incl. inner surfaces when applicable)

Z-buffer (E. Catmull / W. Straber, 1974) Suggested Implementation J. Philip Barnes, 2013Sort QiPs by in(out)-facing and (zc) Initialize: For each pixel (i,j), zref = - For each in(out)-facing QiP (q'gon-in-pyramid)Get pixel box upper-left/lower-right pixel indices For each pixel (i,j) of the pixel boxGet pixel screen coordinates (xp,yp) Test: (p) inside q'gon projection ; OK, interp. (zp)Test at pixel (i,j): zp > zref ; OK, zref = zp Interp. normals at (p) ; shade pixel if it is visible

boundingbox

Ray-casting (Arthur Appel, 1968) Projection-Ray / Z-buffer Hybrid, J. Philip Barnes, 2013Sort QiPs by (zc) ; Initialize: “hit_index” = -1 For each “screen-raybox” pixel (p); zref = - Get pixel sceen x-y ; Cast eye-to-pixel ray For each QiP, test: 2D overlap box includes (p) OK, test: point (p) is inside QiP projectionOK, interpolate (zp) on QiP ; Test: zp > zref OK, hit_index = QiP_index ; zref = zp At hit, interp. normals ; shade pixel if it is visible

pixel box

pixel (p)

x

y

Polygon Pipeline Filter J. Philip Barnes, 2013Initialize: No q'gons-in-pyramid ("QiP"=0) ~ partly/fully Init: “raybox” corners to screen center (if ray-casting) For each quadragon of each polyhedron:Get/set vertex model coords., normals, & in/out colorsModel-to-World transform q'gon vertex coords. onlyCheck: all q'gon vertices pass the 90o exclusion testOK, World-pyramid & perspective x'form vertex coords. Test: q'gon at least partially overlaps the screen ; OK, Model-world and world-pyramid x'form vertex normalsGet 4 corners of q'gon bounding box and 2 of pixel-box Update raybox top-left, low-right corners (if ray-casting)Get q'gon centroid (zc) ; assess in(out)-facing Add q’gon as member of “quadragons in pyramid” (QiP)

RayboxScreen

quadragon

z


x

y

2

3

1

P

PE1P1

E2

E3

Define:Ei Edge vector, vertex(i)-to-next(i+1)Pi Point vector, vertex(i)-to-point (P)z Unit vector out of the page

Observe:(P) inside: E1 x P1 points out of the page(P) outside: E1 x P1 points into the page

Therefore:

Point (P) is outside if, for any vertex (i),( Ei x Pi ) ● k < 0

Postulate: Applies to any convex polygon

Vector determination of "point-in-polygon"

Computer graphics classic problem 3:Given 2D coordinates of vertices and point (P), determine if (P) resides inside



Inter-vertex Interpolation ~ Vector SolutionComputer graphics classic problem 4:Given coords. & properties at vertices 1-2-3, interpolate the properties at point (p)

x

y

p

o

2

3

1

4

Ro4 = Ro1 + R14 = Ro1 + a R1p = Ro2 + R24 = Ro2 + b R23

Finally, interpolate at (p):zp = z1 + [z2 + b (z3-z2) -z1] / a

Equate Ro4 (Dx) & (Dy) to get a, b

Apply vector tip-to-tail addition & scalingSolve for & store two “position parameters”

b

a

/xxyyyyxx/xxyyyyxx

xxyyyyxx

pp

pp

121121

32123212

321321

Classic method: “scan line” (comp. intensive)Contemporary method: Barycentric Coords.Complementary Alternative : Vectors



Constant-curvature Tesselation

• Apply: Breakup curved polygon to avoid edge-rendering artifacts• Whenever at least one vertex is not visible

• Task: Get Cartesian coordinates of curved-polygon edge midpoint• Problem: Barycentric coordinates assume a planar trigon surface

• But the trigon plane typically models a sphere, bowl, cylinder, or saddle surface

• Solution: Assume each trigon edge has constant curvature• Method is exact for spherical surfaces, and perhaps so for cylindrical surfaces

• Unit-normal dot product (ua ●ub) gets cos (q) ; solve for local curve radius (r)

• Vector cross product then gets from (m) to (e), [convex / concave automatic]

Computer graphics classic problem 5:Tesselation of curved polygon Cartesian coordinates

a

be

cf

ga

bem

q

r

Tesselateabc to efg Side view of trigon

edge ab, midpoint (m),true surface point (e),

& unit normals



“Pyranometer Model” of Ambient Lighting

Classic lighting model: Uniform ambient Then add illumination

Problem:“Grey silhouette” until lights are added

Poor foundation of total reflection model

Idea:Vary ambient reflection with orientation

~ good foundation before illumination

Implementation:pyranometer measures "skydome" light

- Rotated on it side ~ half the intensity- Inverted ~ zero intensity (@ albedo=0)

Computer graphics classic problem 6:Model ambient light before illumination

pyranometer disk seesonly the sky "dome"

Sun shade

Define:pyranometer tilt angle (t)Ground albedo (a)

Ambient Reflectance r = 1 - t/p + a t/p



Pyranometer Vs. Uniform Ambient Model (cloudy day)

“Uniform” ambient

“Pyranometer”Ambient

GroundAlbedo = 0.5

GroundAlbedo = 0.0

“Pyranometer”

Ambient


"Render Raptor" ~ Math-modeled Aircraft



Summary ~ Trigons, Normals, and Quadragons

New Solutions: computer graphics classic problems• Complementary, faster, or better, with added insight• Graphical derivation of vector cross product• Fast world-to-pyramid transform ~ just two rotations• Rotation concatenation ~ Review and Renew• 90o exclusion criterion for world-to-pyramid transform• Polygon Pipeline Overview ~ Z-buffer & Ray Casting• Point-in-polygon ~ vector-based determination • Interpolation at point (p) ~ vector-based alternative• Constant-curvature Tesselation• “Pyranometer” ambient-lighting model foundation


Phil Barnes has a Bachelor’s Degree in Mechanical Engineering from the University of Arizona and a Master’s Degree in Aerospace Engineering from Cal Poly Pomona. He has 32-years of experience in performance analysis and computer modeling of aerospace vehicles, engines, and subsystems, primarily at Northrop Grumman. He has authored SAE technical papers on aerodynamics, dynamic soaring, and regenerative soaring. This latest presentation brings together Phil’s knowledge and passions for computer graphics, and geometry math modeling.

About the Author

09 july 2013 update

Documents

d geometry arrays

d computer screen

d screen

d object

geometry definition

pelican aero group

vector operations

philip barnes