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Reconfigurable
Computing Machine
Implementation of
Rational Trigonometry
Algorithms for Missile
Tracking and Prediction
Engineering Excellence Symposium
Azusa, CA
8 November 2006
Richard Wallace
Senior Engineer
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Rational Trigonometry (RT)
What is it?
 RT defines a system of construction for triangles
without reference to circles
 Rational trigonometry does not use any
transcendental values, and is entirely solved
through algebra and quadratic equations.
 It is a recently consolidated form of non-Euclidean
geometry that depends on
 Separation of points (quadrance)
 Separation of lines (spread)
 The name “Rational”
 Calculations use rational numbers or roots of
rational numbers
 Reference Text:
Divine proportions: rational trigonometry to
universal geometry. N.J. Wildberger, Wild Egg,
Australia, 2005, ISBN 097574920X
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Brief introduction to RT (1 of 3)
 Quadrance (Q) is a measure of separation of points, while spread (s) is a
measure of separation of lines. A relation can be shown that quadrance
is the square of distance and spread is the square of the sine of an
angle
2
2
quadrance  distance
spread  sin angle
 The actual definitions of quadrance and spread in RT are independent of
distance, angle, and their trigonometric functions and are based on
finite arithmetic and algebra
 Spread is a dimensionless number between 0 and 1. The spread
between parallel lines is 0. The spread between perpendicular lines is 1.
Its definition can be shown by construction. Given that
l1 and l2 intersect at the point A as shown choose a point
where B  A on one of the lines.
For this construction l1 is used. Let C be the foot of the
perpendicular from B to l2 as shown . If quadrance,
Q(B,C) = Q and Q(A,B) = R the spread s is the ratio:
Q
s  s l1 , l2  
R
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Brief introduction to RT (2 of 3)
A1
Q1
A2
Q1  Q2  Q3 2  2Q 2  Q 2  Q 2 , A1 , A2 , A3 colinear
1
2
3
Q2
Triple Quad
Formula
Q3
A3
A2
Spread Law
For any triangle, A1 A2 A3
with non - zero quadrances
s2
Q3
Q1
A2
A1
Cross Law
s3
s1
Q2
s
s1
s
 2  3
Q1 Q2 Q3
A3
Q3
Q1
A1
c1
Q2
A3
For any triangle, A1 A2 A3 define the
cross c3  1-s3 i.e. non - colinear , then
Q1  Q2  Q3 2  4Q1Q2c3
l1
l2
There are four basic laws
and a restatement of the
Pythagorean theorem in RT
The lines A1A3 and A2 A3 are
perpendicular precisely when Q1  Q2  Q3
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Triple Spread
Formula
s1
s2
For any triangle, A1 A2 A3
s1  s2  s3 2  2s 2  s 2  s 2   4s1s2 s3
1
s3
l3
2
3
Brief introduction to RT (3 of 3)
Given the coordinates of two points (x1,y1) and
(x2,y2), the quadrance between them is:
Q  x2  x1    y2  y1 
2
2
Given the coordinates of two points on each of two
lines (x11,y11), (x12,y12) and (x21,y21) (x22,y22), the spread
between them can be calculated as:
2


x12  x11  y22  y21   x22  x21  y12  y11 
s
x12  x11 2   y12  y11 2 x22  x21 2   y22  y21 2 
2

x1y2  x2 y1 
s
Q1Q2
Spread protractor
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RT projections to spheres (1 of 2)
Rational Trigonometry has spherical projections.
Given the sphere x2 + y2 + z2 = 1 and center O = [0, 0, 0] any two non-antipodal
points A and B lying on it determine a unique spherical line which is the
intersection of the sphere with the plane OAB. Any two such spherical lines
intersect at a pair of antipodal points.
As shown on the left, a spherical triangle is formed by three spherical points
A, B, and C and three spherical lines, and on the right the corresponding
projective triangle, consisting of three projective points a, b, and c and the
three projective lines that they form.
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RT projections to spheres (2 of 2)
q
 Projective Thales’ theorem S 
Spread S; Quadrances q & r as shown
r
 There are twelve other theorems
which we don’t have time to go
through. Please reference the text
 Summary: The geometry is complete
to cover conic, spherical, and elliptic
constructions as well as planar
RT does not cover circular or harmonic
functions to deal with circular motion,
Fourier analysis and the like, but those wavelike functions with no natural zero would be
better not called “trigonometric.” They are
not related to triangles.
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Classic Thales’ theorem:
If A, B and C are points on a
circle where the line AC is a
diameter of the circle, then
the angle ABC is a right
angle.
ECF in Euclidian (ET) and Rational form
z
P
R
q
O
s
y
Where
geocentric latitude
 = longitude
Where
and
x = r cos cos 
y = r cos sin 
z = r sin 
Therefore, a unit vector along r is given by
cos  cos 
r̂  cos  sin   
sin 


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
s  x2 x2  y2

q  x2  y2
r = vector from earth center to point P
r = magnitude of vector r
P12
x

 x
2
R  x2  y2  z 2
and
x 2  sqR
y 2  1  s qR
z 2  1  q R

sqR 


Pˆ   1  s qR 


 1  q R 
 y2  z2

RT method ½ of the solution
 Questions:


Why not use CORDIC algorithms?
Isn’t rational trigonometry is just a reformulation of what we already
know in Euclidian geometry?
 Not quite…


CORDIC is an approximation for non-HW multiplier FPGAs. We use
modern FPGAs. Slower computational speed issues using CORDIC
RT does expresses the planar and solid geometry we know with
three improvements:
 The calculations are exact rather than approximate, eliminating
compounded error
 Calculations can use more efficient fixed-point, algebraic
operations taking advantage of HPC/FPGA structures/circuits
 Few, or no, transcendental calculations will result in faster time
to solution and fewer computational resources needed for such
calculations.
 The other ½ is the machine implementation…
COordinate Rotation DIgital Computer
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Reconfigurable Computing (RC)
Prototype System
 Elements
 Xilinx Virtex-4 XC4VLX160 or
XC4VSX55 device
 Co-processing with AMD Opteron
 All programmable using Celoxica DK
 HTX Interface
 Data transfer up to 3.2GB/s
 Direct access to entire host system
memory space
 Bridge FPGA manages I/O tasks
freeing user FPGA for co-processing
 Dedicated Memory
 24MB QDR SRAM on board
 9.6 GB/sec max transfer rates
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RCHTX-XV4
Programming the RC prototype
Use the appropriate DK design
suite components; not all
elements needed
Refactor existing C/C++
algorithms in Handel-C
Use Dr. Prasanna, et. al. (USC)
method to improve algorithms
Celoxica full DK design suite
Celoxica DK
subset needed
Dr. V. Prasanna, USC
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Where RT and RC synergize
 Computational processing made of highly flexible computing
“fabrics” of single-purposed circuits that can be reconfigured,
reconnected, into new-single purposed circuits using control
configurations driven by the computational calculation itself.
 The principal difference of reconfigurable computing (RC)
when compared to using ordinary microprocessors is the
ability to make substantial changes to the data path itself in
addition to the control flow of the computation.
 No instructions
 Hardware is configured for a
particular application
 Parallelism
 Multiple functional units of a
given type
 Better resource utilization
than general purpose
processor
 Can be reconfigured for
new application
 Memory structure tailored to
the application
 Good for data-intensive
applications
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Reconfigurable
Computing Fabric
Fixed construction
von Neumann machine
Rational Trigonometry (RT)
Why use it?
 What we know
 Knowledge that the majority of engagement angles are
acute
 Knowledge that the majority of tracking calculations can be
represented in fixed-point
 Knowledge that the majority of filter calculations are scalar
and the calculations can be algebraic rather than
trigonometric
 Knowledge that fixed-point and scalar operations are best fit
to structural calculation rather than temporal calculation in
RC devices
 What we get
 With RT + RC  Accuracy, Precision, and Speed
 Flexibility, Size, Weight, Power, and Thermal efficiencies
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Progress & Improvements
 Progress
 Testing ET to RT mapped algorithms on GP system. Promising
results, > 20% improvement on GP
 Discrete filter operations using Bridge as controller
 System thermal improvements expected based on code written
USC method
Prasanna, et. al.
FPGA register Energy Use
 Challenges
 Assuring all operations are RT, not ET
 Proper use of Handel-C pragmas
 Balancing use of RC resources
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Questions and Answers
Questions?
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