1 Pelican Aero Group J. Philip Barnes 07 Sept 2012 update Trigons, Normals, and Quadragons...

1 Pelican Aero Group

J. Philip Barnes

07 Sept 2012 update

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

Problems of Computer Graphics

Notes / Options:Point-by-point slideshow F5 keyDon't miss the slide footnotes


Trigons, Normals, and Quadragons J. Philip Barnes 27 May 2012 www.HowFliesTheAlbatross.com

Overview

Presentation Contents • “Flexible-cylinder” geometry grid • Geometry math modeling tools• Vectors ~ history & key operations • Fast world-to-screen transformation• Review/Renew: Z-buffer & Ray-casting • Vector solution : “Point-in-polygon” • Vector solution: Inter-vertex interpolation• Vector solution: Tesselate Cartesian coords.• “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

ql

q

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



“TrigonoSoar” Math-modeled 3D Surface

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

x

z

y



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

AyBxAzBy

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

cross product components as shown at upper right



Fast World-to-screen Transform & Screen-cone Criterion

f

y

y

y-ye

z

z-ze

1. cosc = (ze-zf) / Ref ; sin y = (xf -xe) / (Ref cosc) 2. Rotate (y) about y-ye axis through point (e)3. Rotate (c) about x-xe axis through point (e)4. Optionally rotate (z) for “cockpit-roll” effect5. Project to screen (perspective transformation)

v

fw

e

xs, zs

xp

yp

c

c

x-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).First test new "Screen-cone" criterion. Skip projection for any quadragon with all vertices (v) which after transform would remain outside the screen cone defined by the corner angle (s). Implement exclusion via the dot product of the

eye-to-vertex vector (ev) and focus vector (ef),

testing all four vertices of each quadragon:

ev ef / { | ev || ef | } < cos s

vw

Screencone

s FocuspointEye

point

vertex

Ref

After transform,model-to-world

Projected toscreen plane

modelaxes

After world

rotations

vm



Screen Plane

Polygon Pipeline Optimization (incl. inner surfaces when applicable)

Z-buffer (E. Catmull / W. Straber, 1974) Optimized Implementation by P. Barnes, 2011Sort 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, P. Barnes, 2011Sort 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 P. Barnes, 2011

Initialize: 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-World (M-W) x'form q'gon vertex coords. onlyTest: Will any of 4 vertices project in the screen cone?OK, World-rotate & screen-project q'gon vertex coords. Test: q'gon at least partially overlaps the screen ; OK, M-W transform & World-rotate q'gon 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)

Raybox

Screen Cone

Screen

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 Solution

Computer 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”

/xxyyyyxx

/xxyyyyxx

xxyyyyxx

pp

pp

121121

32123212

321321

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



Accurate Tesselation of Cartesian Coordinates

• 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• 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:Accurate tesselation of curved polygon Cartesian coordinates

a

b

e

cf

ga

be

m

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



TrigonoSoar ~ All Methods Herein Applied

Uniform Ambient

“pyranometer”Ambient Model

GroundAlbedo = 0.5

GroundAlbedo = 0.0



"Render Raptor" ~ Math-modeled Aircraft



Summary ~ Trigons, Normals, and Quadragons

New Solutions: computer graphics classic problems

• Complementary, faster, or better, with added insight

• World-to-screen transformation ~ just two rotations

• "Screen-cone" world-to-screen exclusion criterion

• Point-in-polygon ~ vector-based alternative

• Interpolation at point (p) ~ vector-based alternative

• “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



Vector-based World-to-screen Transformation

f

y-ye

z

z-ze

v

E xp

yp

x-xe

FocuspointEye

point

After transform,model-to-worldmodel

axes

After world

rotations

e

Optional "cockpit-roll"

Ev

Ef

e

G

Screen vector:|G| = (ze-zs) tan exp=G ● i ; yp=G ● j

Where, for unitvectors, e≡E/|E|:cos e = ev

● ef

sin e = | ev x ef |

s

vertex

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