Requirements to a Line Smoothing Method

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Requirements to a Line Smoothing Method

• Although it is a string of line segments, the
  visual impression of it should be a smooth line
• Not only the pieces of a line between the given
  adjacent points should be connected, the
  derivatives of at least the first order must be
  continuous at the junction points
• The figures of the lines on the two sides of a
  junction point and near to it should be symmetric
  
• Be able to avoid self-intersection as much as
  possible

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Piecewise Line Smoothing Using Third Order
Polynomials with Derivatives from Five Points
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Third Order Polynomials with Derivatives from
Five Points (1)

• The basic principle of this method is to set up a
  third order polynomial equation for a curve
  between two successive points, and it is demanded
  that the whole line has the continuous first
  order derivatives to insure the smoothness of the
  line.
• The first order derivatives on each of the two
  successive points is determined by this point
  plus last two points and next two points (5
  points altogether).

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Third Order Polynomials with Derivatives from
Five Points (2)

• The stipulated geometric condition for the
  tangential line passing through the middle point

P1
P5
P2
P4
P3
C
A
D
B
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Third Order Polynomials with Derivatives from
Five Points (3)

• Through mathematical derivation, the derivative
  t3 on the point P3 can be expressed as

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Third Order Polynomials with Derivatives from
Five Points (4)
(a02b02)1/2

• In order to be convenient, we use cos?3 and
  sin?3 for t3 (i.e., tg?3), and the numerator and
  the denominator of the expression for t3 are seen
  as the two edges of a right triangle

b0
?
a0
When w2w30, cos ?3 and sin ?3 are uncertain,
let w2w31
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Third Order Polynomials with Derivatives from
Five Points (5)
Let the third order polynomial equations between
the two successive points Pi and Pi1 be
xp0p1zp2z2p3z3 yq0q1zq2z2q3z3
In these equations, pi, qi (i0,1,2,3) are
constants, z is the parameter, when the curve is
being generated from Pi (xi, yi) to Pi1 (xi1,
yi1), z is changing from 0 to 1.
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Third Order Polynomials with Derivatives from
Five Points (6)
These equations should satisfy the following
conditions

When z0, xxi, yyi, dx/dzrcos?i,
dy/dzrsin?i, When z1, xxi1, yyi1,
dx/dzrcos?i1, dy/dzrsin?i1. Where
r(xi1-xi)2(yi1-yi)21/2
Pi1
?i
Pi
?i1
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Third Order Polynomials with Derivatives from
Five Points (7)

• From these conditions, pi and qi (i0,1,2,3) can
  be uniquely defined
• p0xi
• p1rcos?i
• p23(xi1-xi)-r(cos?i12cos?i)
• p3-2(xi1-xi)r(cos?i1cos?i)
• q0yi
• q1rsin?i
• q23(yi1-yi)-r(sin?i12sin?i)
• q3-2(yi1-yi)r(sin?i1sin?i)

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Third Order Polynomials with Derivatives from
Five Points (8)

• Supplement of two points on each end for an open
  line
• Suppose that the first three given points
  (x3,y3), (x4, y4), (x5, y5), and the two points
  to be supplied (x2,y2) and (x1,y1) are all on the
  following line
• xg0g1zg2z2
• yh0h1zh2z2

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Third Order Polynomials with Derivatives from
Five Points (9)

• where gk, hk (k0,1,2) are constants, z is the
  parameter, and suppose that when zj, xxj, yyj
  (j1,2,3,4,5), then
• x23x3-3x4x5
• x13x2-3x3x4
• y23y3-3y4y5
• y13y2-3y3y4

The supplement of the points after the last given
point is similar to this.
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Third Order Polynomials with Derivatives from
Five Points (10)

• Advantages
• To be strict mathematically
• Relatively simple in calculation
• The resulted curve can pass through every given
  point exactly while has the continuous first
  order derivatives everywhere on the line
• When given points are relatively dense, the
  resulted curve will be satifactory

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Third Order Polynomials with Derivatives from
Five Points (11)

• Disadvantages
• The graphic representation would not be ideal at
  the place of the sudden turning of the curve
• For the successive zigzag meanders, the curve
  sometimes intersects with itself

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Piecewise Line Smoothing Using Third Order
Polynomials with Derivatives from Three Points
15
Third Order Polynomials with Derivatives from
Three Points (1)

• The basic principle of this method is similar
  with the method used in the piecewise line
  interpolation using the three order polynomial
  with derivatives from five points. The main
  difference lies in the method of determination
  for the derivatives.

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Third Order Polynomials with Derivatives from
Three Points (2)

• The derivative on every given point is determined
  by the relative positions of the last and next
  points to this point. Therefore, three successive
  points are used to calculate the derivative of
  the curve on the middle point
• Let the three points be Pi-1, Pi, Pi1, their
  coordinates are (xi-1, yi-1), (xi, yi), (xi1,
  yi1) respectively

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Third Order Polynomials with Derivatives from
Three Points (3)
Suppose that there is a point M (xm, ym) on the
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Third Order Polynomials with Derivatives from
Three Points (4)
Y
Pi
E
Ti
F
ai
Pi2
Pi-1
M
ai1
Pi1
Ti1
O
X
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Third Order Polynomials with Derivatives from
Three Points (5)

• The slope of the line segment PiM is
• KPiM (ym-yi) / (xm-xi)
• therefore, the slope of the tangential line on
  point Pi is
• KEF -1 / KPiM
• Let the direction of EF be identical with the
  moving direction of the curve.

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Third Order Polynomials with Derivatives from
Three Points (6)

• After the quadrant judgement, the angle abetween
  the directed line EF and the X-axis can be
  uniquely determined within the range of 0 - 2p.
  tgais the first order derivative at the point Pi
  on the curve.
• All the derivatives tgai (i2,3,...,N-1) can be
  calculated in this way.

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Third Order Polynomials with Derivatives from
Three Points (7)

• The establishment of the parametric equations of
  the third order polynomials is just like the way
  for setting up the equations for the third order
  polynomials with derivatives from five points.

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Third Order Polynomials with Derivatives from
Three Points (8)

• Attention must be paid to
• For an open line, the derivative of the first or
  last point can be seen as the slope of the
  tangential line on the first or last point of a
  circle or a parabolic line which passes through
  the first three points or the last three points
• For special cases such as three successive points
  locate on a same straight line, or the slope of
  the line segment KPiM is nearly infinitive,
  judgement and assignment must be made in advance

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Third Order Polynomials with Derivatives from
Three Points (9)

• r can be changed in order to change the
  tightness of the curve. Therefore, in order to
  regulate the tightness of the curve in time, the
  two angles which are formed by the string
  Pi-1PiPi1Pi2 are used as the components of the
  weight function to regulate the value of r,
  called r
• r r (1-z)sinßizsinßi1

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Third Order Polynomials with Derivatives from
Three Points (10)
ßi
ßi1
r r (1-z)sinßizsinßi1
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Third Order Polynomials with Derivatives from
Three Points (11)

• Advantages
• This method can give satisfactory result even if
  the given points are distributed relatively
  sparse
• The curve passes through all the given points
• The first order derivatives are continuous at
  everywhere on the curve
• Graphical symmetry near every point
• The ability to avoid self intersection
• To be simple in mathematic principle and easy to
  write the program

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Third Order Polynomials with Derivatives from
Three Points (12)
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Questions for Review (1)

• What are the general requirements for line
  smoothing?
• What are the advantages and disadvantages of
  weighted average of normal axis parabolic lines?
• What are the differences between the methods of
  piecewise line smoothing using third order
  polynomials with derivatives from five and from
  three points?

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Questions for Review (2)

• Please describe the concrete steps of line
  smoothing using third order polynomials with
  derivatives from three points.
• How are the derivatives calculated using the
  five point method?
• How are the derivatives calculated using the
  three point method?
• What are the frame conditions with respect to the
  change of parameter z in the third order
  polynomial line smoothing?