Analysis of Newton’s Method in Draw-CAD
Naman Agarwal
May 4, 2012
under guidance of,
Prof. Abhiram Ranade,
IIT Bombay
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
DrawCAD-Introduction
DrawCAD - a free hand drawing tool being developed at IIT
Bombay
Allows a user to input geometric figures in a free-hand sketch
format
A user is allowed to input constraints like equality of lines,
parallelism, perpendicularity etc.
These constraints are entered sequentially and are solved as
and when they are entered.
The solution methodology used is Newton’s Method for
Multiple Equations in Multiple Dimensions.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Problem Description
In this presentation we want to explore the convergence
properties of Newton’s method in the setting of Draw-CAD’s
geometric constraints and also apart from convergence we
want to know when the convergence happens to
Non-degenerate solutions.
By non-degenerate solutions we mean that if a user has drawn
two points seperately, they must remain seperate in the final
solution too. For example when a drawn line shrinks to one of
0 length in the solution, it is considered a degenerate solution.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Outline
Newton Method’s Step Description
Geometric Interpretation of the Newton’s Method Iteration.
Convergence properties and proofs in different cases Single Constraint Scenarios involving Distance Equality,
Perpendicularity and Parallelism
Mulitple Distance Equality Constraints involving Disjoint Lines.
Point Shared between two lines. Wedge Case
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Notation
Newton Step
Moore Penrose PseudoInverse
Notation
The state of a geometric system is captured by a column
vector represented as X. The entries in X ∈ Rn correspond to
the coordinates of the points of the system. Notationally we
assume X to be arranged in the order that the y coordinates
are written first before the x coordinates.
Example
Consider a simple case of a user drawing two lines l1 and l2 with
end-points (x1 , y1 ), (x2 , y2 ) and (x3 , y3 ), (x4 , y4 ) respectively. The
state of this Geometric System would typically be represented by a
column vector(X) as
T
X = y 1 y 2 y 3 y 4 x1 x2 x3 x4
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
(1)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Notation
Newton Step
Moore Penrose PseudoInverse
Notation Continued
Let C be the set of constraints applied on a system. The error
function f is a function from Rn → RkC k .
Aim is to find an X such f (X ) = 0
Also, J(f (X )) ∈ RkC k×n represents the jacobian of the
function f at point X, with each row representing the gradient
vector of the particular constraint in state X.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Notation
Newton Step
Moore Penrose PseudoInverse
Newton-Step
The Newton Step Equation at the n’th step is given by
f (Xn+1 ) − f (Xn ) = 0 − f (Xn ) = J(f (Xn ))(Xn+1 − Xn )
We solve the above system of equations by using the Moore
Penrose pseudo inverse.
Xn+1 = Xn − J + ((f (Xn )))f (Xn )
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Notation
Newton Step
Moore Penrose PseudoInverse
Pseudo-Inverse
For A an m × n matrix, A+ is an n × m matrix satisfying the
following properties AA+ A = A
A+ AA+ = A+
(AA+ )∗ = AA+
(A+ A)∗ = A+ A
Following properties help us
The Moore Penrose pseudo-inverse always exists and is unique
for any A
Given a linear system B = AX , X = A+ B, is the solution with
the lowest norm
1
T
The pseudo-inverse of a row vector A is kAk
2 ∗ A
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Constraint Equations
Through the presentation we shall focus on three most used
constraints distance equality, perpendicularity and parallelism
constraints single constraint settings. The error equations for
these constraints are given by For Distance Equality (y2 − y1 )2 + (x2 − x1 )2 − (y3 − y4 )2 − (x3 − x4 )2 = 0
For Perpendicularity (y2 − y1 ) ∗ (y3 − y4 ) + (x2 − x1 ) ∗ (x3 − x4 ) = 0
For Parallelism - (y3 − y4 ) ∗ (x2 − x1 ) − (x3 − x4 ) ∗ (y1 − y2 ) = 0
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Geomteric Interpretation
The newton’s method step is given by the following equation J(f (X )) ∗ ∆X = −1 ∗ f (X )
(2)
∆X represents the change in the solution guess for the next
iterate. Movement Mi of the point is the subvector
corresponding to the point’s x,y coordinates in the iteration.
Parts of the gradient of a particular function corresponding to
a point’s x,y coordinates can be seperated out into a seperate
vectors which we refer to as projected-gradients pGrad at
those points.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Substituting values from the previous slide, the newton’s
equation method now reads

  
J1 (f (X )).∆X
0
J2 (f (X )).∆X  0

  

 .
.

= 

 .
.

  

 .
.
Jn (f (X )).∆X
Naman Agarwal
e
Analysis of Newton’s Method in Draw-CAD
(3)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Geometric Interpretation
We know that
Ji (f (X )).∆X =
X
pGradi (X ).Mi (X )
(4)
pointsPi
Picking ∆ of smallest norm means that we are picking a
solution with minimum root square movement. i.e.
X
∆X =
kMi (X )k2
(5)
pointsPi
Find such movements of the points which satisfy the required
condition with the projected-gradients as well as minimise the
sum total of the movements.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Gemometric Interpretation - Distance Equality
Distance Equality - The gradient of the error function comes
out to be


−2 ∗ (y1 − y2 )
−2 ∗ (y2 − y1 )


 2 ∗ (y3 − y4 ) 


 2 ∗ (y4 − y3 ) 

C =
(6)
−2 ∗ (x1 − x2 ) .


−2 ∗ (x2 − x1 )


 2 ∗ (x3 − x4 ) 
2 ∗ (x4 − x3 )
Projected-gradient at each point points along the direction of
the line on which the point is.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Gemometric Interpretation - Perpendicularity
Perpendicularity - The gradient of the error function comes
out to be


(y3 − y4 )
(y4 − y3 )


(y1 − y2 )


(y2 − y1 )

C =
(7)
(x3 − x4 ) .


(x4 − x3 )


(x1 − x2 )
(x2 − x1 )
Projected-gradient at a point points along the direction of the
other line.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Gemometric Interpretation - Parallelism
Parallelism - The Projected-Gradient of the error function
comes out to be


−1 ∗ (x3 − x4 )
−1 ∗ (x4 − x3 )


 (x1 − x2 ) 


 (x2 − x1 ) 

C=
 (y3 − y4 )  .(8)


 (y4 − y3 ) 


−1 ∗ (y1 − y2 )
−1 ∗ (y2 − y1 )
Projected-gradient at a point points along the direction
perpendicular to the other line.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
The movements are pictorially depicted in the following figure -
Figure : Figure depicting the gradients and movements in case of specific
constraints
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Perpendicularity
Both the lines move towards each other if the angle between
them is obtuse and move away from each other if the angle is
acute
The error reduces at least by a factor of four on every
iteration.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof
Solution vector T
X = y1 y2 y3 y4 x1 x2 x3 x4
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof
Solution vector T
X = y1 y2 y3 y4 x1 x2 x3 x4
The perpendicularity constraint (y3 − y4 )(y1 − y2 ) + (x1 − x2 )(x3 − y4 )
(9)
The Jacobian for the above constraint is given by the row
matrix J(X ) where
y3 − y4 y4 − y3 y1 − y2 y2 − y1 x3 − x4
J(X ) =
(10)
x4 − x3 x1 − x2 x2 − x1
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Therefore the new point X ∗ is given by
X ∗ = X + −1 ∗ J + (f (X )) ∗ f (X )
T
= X − J (f (X )) ∗ f (X )/kJ(f (X ))k
Naman Agarwal
(11)
2
Analysis of Newton’s Method in Draw-CAD
(12)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Therefore the new point X ∗ is given by
X ∗ = X + −1 ∗ J + (f (X )) ∗ f (X )
T
= X − J (f (X )) ∗ f (X )/kJ(f (X ))k
Let t(X ) =
(11)
2
f (X )/kJ(f (X ))k2
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
(12)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Therefore the new point X ∗ is given by
X ∗ = X + −1 ∗ J + (f (X )) ∗ f (X )
T
= X − J (f (X )) ∗ f (X )/kJ(f (X ))k
(11)
2
f (X )/kJ(f (X ))k2
(12)
Let t(X ) =
Substituting the corresponding values for X and the Jacobian
we get


y1 − (y3 − y4 )t
y2 + (y3 − y4 )t 


y3 − (y1 − y2 )t 


y4 + (y1 − y2 )t 
∗


X =
(13)

x
−
(x
−
x
)t
1
3
4


x2 + (x3 − x4 )t 


x3 − (x1 − x2 )t 
x4 + (x1 − x2 )t
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Let,
a = y1 − y2
b = y3 − y4
c = x1 − x2
d = x3 − x4
Now consider t.
t = (a ∗ b + c ∗ d)/2 ∗ (a2 + b 2 + c 2 + d 2 )
(14)
(15)
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Let,
a = y1 − y2
b = y3 − y4
c = x1 − x2
d = x3 − x4
Now consider t.
t = (a ∗ b + c ∗ d)/2 ∗ (a2 + b 2 + c 2 + d 2 )
(14)
(15)
AM ≥ GM inequality implies that
(a2 + b 2 + c 2 + d 2 ) >= 2(ab + cd). Therefore t ≤ 1/4
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Consider the new error, Errornew
Errornew = (a − 2tb)(b − 2ta) + (c − 2td)(d − 2tc)
2
2
2
2
(16)
2
= (ab + cd)(1 + 4t ) − 2t(a + b + c + d )
(17)
= (ab + cd)(4 ∗ t 2 )
(a ∗ b + c ∗ d)
= Errorold ( 2
)2
(a + b 2 + c 2 + d 2 )
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
(18)
(19)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Perpendicularity-Degeneracy
Degeneracy - If the lines start out parallel to each other, there
will not be a degenrate solution possible
Non-degeneracy - The perpendicular component of one line
on the other goes on to continuously increase except for when
the lines are parallel or perpendicular. This gives
non-degeneracy in the solution.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof of Degeneracy
For two lines l1 and l2 with endpoints (x1 , y1 ), (x2 , y2 ) and
(x3 , y3 ), (x4 , y4 ) the perpendicular component of l1 on l2
(l1 ⊥ l2 ) is given by l1 ⊥ l2 = k
(x3 − x4 )(y1 − y2 ) + (y3 − y4 )(x1 − x2 )
k
(x3 − x4 )2 + (y3 − y4 )2
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof of Degeneracy
For two lines l1 and l2 with endpoints (x1 , y1 ), (x2 , y2 ) and
(x3 , y3 ), (x4 , y4 ) the perpendicular component of l1 on l2
(l1 ⊥ l2 ) is given by l1 ⊥ l2 = k
(x3 − x4 )(y1 − y2 ) + (y3 − y4 )(x1 − x2 )
k
(x3 − x4 )2 + (y3 − y4 )2
Borrowing the notation from the convergence proof we have a = y1 − y2 , b = y3 − y4
c = x1 − x2 , d = x3 − x4
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof of Degeneracy
For two lines l1 and l2 with endpoints (x1 , y1 ), (x2 , y2 ) and
(x3 , y3 ), (x4 , y4 ) the perpendicular component of l1 on l2
(l1 ⊥ l2 ) is given by l1 ⊥ l2 = k
(x3 − x4 )(y1 − y2 ) + (y3 − y4 )(x1 − x2 )
k
(x3 − x4 )2 + (y3 − y4 )2
Borrowing the notation from the convergence proof we have a = y1 − y2 , b = y3 − y4
c = x1 − x2 , d = x3 − x4
Therefore,
a∗d −b∗c
l1 ⊥ l2 = k √
k
b2 + d 2
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof of Degeneracy
Also from the convergence proof for the next iteration, Let
the new values of the above quantities be a’,b’,c’,d’. We now
have,
a0 = a − 2bt, b 0 = b − 2at
c 0 = c − 2dt, d 0 = d − 2ct
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Proof of Degeneracy
Also from the convergence proof for the next iteration, Let
the new values of the above quantities be a’,b’,c’,d’. We now
have,
a0 = a − 2bt, b 0 = b − 2at
c 0 = c − 2dt, d 0 = d − 2ct
Plugging these values into the expression for perpendicular
components and simplifying, we get
(l1 ⊥ l2 )new = k p
(ad − bc)(1 − 4t 2 )
(b 2 + d 2 )(1 − 8t 2 ) − 4t 2 (a2 + c 2 )
k (20)
which can be seen to be greater than the previous value
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
ad−bc
a2 +c 2
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Parallelism
Both the lines move away from each other if the angle
between them is obtuse and move towards from each other if
the angle is acute.
The error function comes out to be
Errornew = Errorold (
(c ∗ b − a ∗ d)
)2
+ b2 + c 2 + d 2 )
(a2
The error reduces at least by a factor of four on every
iteration.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
(21)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Parallelism-Degeneracy
Degeneracy - If the lines start out perpendicular to each other,
there will not be a degenrate solution possible
Non-degeneracy - The perpendicular component on one line
on the other goes on to continuously increase except for when
the lines are parallel or perpendicular. This gives
non-degeneracy in the solution.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Distance Equality
One line constantly increases in length while the other
decreases and stay parallel to their original directions.
The error function comes out to be
Errornew = Errorold ∗ ((a2 − b 2 )/(a2 + b 2 ))2
The error reduces at least by a factor of four on every
iteration.
Always gives a non-degenerate solution as one line always
increases in length.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
(22)
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Multiple Disjoint Lines
Distance Equality Constraints divide the system into
equivalence classes. Only two of these affected by the new
constraint.
Each line shrinks or expands symmetrically from both ends.
Each equivalence class shrinks and expands equally, thereby
maintaining the previous constraints.
The difference between the lengths of the two equivalence
class is given by
(kl2 k − kl1 k) ∗ (1/2) ∗ (kl2 k − kl1 k) ∗ (t ∗ kl2 k − k ∗ kl1 k)
(t ∗ kl2 k2 + k ∗ kl1 k2 )
(23)
The above sequence does go onto converge to 0
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Simple perpendicularity constraint on two disjoint lines
Simple parallelism constraint on two disjoint lines
Simple distance equality constraint on two disjoint lines
Mulitple Disjoint Lines linked with Multiple Distance Equality Const
Case of a shared point - the wedge case
Point Sharing Case - Wedge
The common point moves in the direction which is the sum of
the movements caused by the two constraints.
Similarly in this case also we have one line increasing and the
other decreasing in length, and the error function decreases on
every iteration.
Such a simple addition of movements does not occur the
same point is shared over two constraints.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD
DrawCAD-Introduction
Outline
Newton’s Method Description
Geometric Interpretation of the Newton’s Method Step
Convergence Analysis in specific settings
Conclusion and Further Work
Conclusion and Further Work
We were able to show for simple cases the convergence to
non-degenerate solutions. These ideas need to be extended to
more complex scenarios, possibly be genralised over all
systems having at least one valid solution.
The structure of the Jacobian is hard to predict in systems
with more than one constraint. Hence analyzing the system
algebraically seems difficult.
Use of Geometric properties, gives structure and more work
can possibly be done in generalizing the approach to more
complex cases. However this approach seems to be specific to
cases and gradient directions.
Naman Agarwal
Analysis of Newton’s Method in Draw-CAD