L5 – Curves in GIS

NGEN06 & TEK230: Algorithms in Geographical Information Systems

by: Sadegh Jamali (source: Lecture notes in GIS, Lars Harrie)1

L5- Curves in GIS

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L5- Curves in GIS

Aim

… to provide a background to how curves can be used to represent line objects in GIS.

Content

1. Overview

2. Interpolation of mathematical functions

3. Piecewise polynomials

4. Hermite interpolation

5. Cubic spline

6. Curves in the plane

7. Curves in Geography Markup Language (GML)

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L5- Curves in GIS

Overview

Most mapping software handles vector data as line strings (i.e. straight line segments between vertices).

Fig. a line string

Drawback: too angular!

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L5- Curves in GIS

Overview

Limitation of the solid curve: we can only describe curves with increasing x-values.

Solid: a curve with allowed mathematical functiony=f(x)

Dashed: a general curve in the plane

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L5- Curves in GIS

Interpolation of mathematical functions

Fitting a polynomial:

In general, a polynomial of degree n can be adapted to pass through n+1 points.

How to compute the unknown parameters ()?

the polynomial must pass through ALL the points.

A polynomial of degree n is written as:

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L5- Curves in GIS

Example: create a 3rd order polynomial that passes through the 4 points in the table.

𝐴 .𝑋=𝐿

Non-singular system:

Singular system

𝑋=¿

Linear system of equations:

3rd order polynomial

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L5- Curves in GIS

The fitted polynomial of 3rd degree to the points

The points

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L5- Curves in GIS

Problems in adapting polynomials of high degrees

Red curve: Runge function

Blue curve: a 5th-order interpolating polynomial (using six equally spaced interpolating points)

Green curve: a 9th-order interpolating polynomial (using ten equally spaced interpolating points).

Example: approximate the Runge’s function (between -1 and +1) using polynomials

Runge’s function

Runge’s fucntion

degree= 9

degree= 5

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L5- Curves in GIS

Runge’s phenomenon

Error=0; at the interpolating points

Error >>0; between the interpolating points (especially in the region close to the endpoints 1 and −1 for higher-order polynomials.

Problems at the ends of the area or fluctuating severely in the border areas of the interval

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L5- Curves in GIS

Piecewise polynomials

As seen, if we want the polynomial to pass many points we have to use a polynomial with high degree, but that will provide undesired effects especially in the border areas of the interval.

Solution? using several piecewise polynomials with lower degrees

Figure: a curve constituted of three polynomials (normally degree 3) between each two following points.

1st polynomial

2nd polynomial

3rd polynomial

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L5- Curves in GIS

The piecewise curves are always continuous within themselves.

But the problem here is how to get good properties at the connection points

between the polynomials.

the curves that guarantee that the first order derivatives(and sometimes also the second order derivatives) are continuous also at the connection points.

Solution? Hermite interpolation and cubic splines:

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L5- Curves in GIS

Hermite interpolation

… based on interpolation with piecewise polynomials (3rd degree).

Requirements:

1) the interpolated curve must pass through all points (each 2 following points)

2) the first order derivative must be continuous EVERYWHERE.

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L5- Curves in GIS

Polynomial in Hermite interpolation:

Figure: parameters in Hermite interpolation

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L5- Curves in GIS

Question: dose this polynomial really satisfy the requirements ?

Requirements:

1) the interpolated curve must pass through all points (each 2 points)

If we apply this to the first polynomial (): we get that at the start point () and at the end point of the first polynomial ().

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L5- Curves in GIS

Requirements:

2) the first order derivative must be continuous everywhere.

The derivative of with respect to :

This function is continuous within a polynomial. Question: but what happens between the polynomials?

the derivatives at the start and end points:

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L5- Curves in GIS

Be aware that the result of the derivative with respect of t contain an h factor, which is the scaling factor between t and x. This implies that the derivative with respect to x is equal to k. That is we obtain for the first

polynomial (in the Figure) that the derivative at the start point is k1 and that

the derivative at the end point is k2, which is correct.

So all the constraints are satisfied and, hence, the

equation provides the Hermite interpolation.

L5- Curves in GIS

Example: create Hermite interpolation of the points and first order

derivatives in the given Table

Figure: Hermite interpolation: the solid line is the curve for the first order derivative (der1) and the dashed line is the curve for the first order derivative (der2).

soliddashed

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L5- Curves in GIS

Calculation of derivatives at the points:

m = the number of points (here m =4). The first and last derivatives are set by using the slope to the nearest point.

Note: the derivatives der2 in the table were derived using the above method.

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A standard method is to use the following differences:

L5- Curves in GIS

Cubic spline

A piecewise 3rd order polynomial; a type of Hermite interpolation with additional requirements that the second order derivatives are continuous everywhere.

For a curve with m points:m-1 piecewise cubic polynomials 4(m-1) unknown parameterspassing the start and end points in all the intervals 2(m-1) constraints Hermite’s requirement for the first order derivatives m-2 constraints requirement for the second order derivatives m-2 constraints

Degree of freedom: 4(m-1) - ( 2(m-1) + (m-2)+ (m-2)) = 2

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L5- Curves in GIS

Cubic spline

Ways of setting the two last constraints?

Clamped cubic spline: the first order derivatives at the start and end points of the whole curve are given.

Natural cubic spline: the second order derivatives are zero at the start and end points of the whole curve.

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L5- Curves in GIS

How to compute natural cubic splines?

1) compute values for the first order derivatives () by solving the following linear system equations.

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L5- Curves in GIS

2) apply the Hermite interpolation function

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L5- Curves in GIS

Example: compute the natural cubic splines for the points given in previous example

Note: the curve is smoother than the other Hermite interpolations due to the requirement of continuous second order derivatives

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Figure: Natural cubic spline interpolation of the points. The first order derivatives at the points are given in the table above.

L5- Curves in GIS

Curves in the planeThis needs a more general expression than interpolations methods because x-values can decrease too

Parametric representation x=x(t) and y=y(t) where t is a local variable. Normally t goes from 0 to 1.

Figure: Parametric representation of a curve in the plane. 24

L5- Curves in GIS

General expressions for the curve the curves in the plane are represented using cubic polynomials

Matrix form:

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L5- Curves in GIS

Hermite curves All of the properties of the Hermite interpolation are also valid for the Hermite curves.

Hermite curves in Matrix form:

Where Pi are the end pointsRi are the tangent vectors of the same curve

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L5- Curves in GIS

Figure: Hermite curves in the plane for two end points (dots) but with different tangent vectors (arrows).

Example: Hermite curves in the plane

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L5- Curves in GIS

Bezier curves … are similar to the Hermite curves but with known directions for the curve at the start and end points.

where P1 and P4 are the end points of the curves, and P2 and P3 are points that show the direction of the curves at the end points

Figure: Bezier curves in the plane. 28

L5- Curves in GIS

Spline curves in the planeThe mathematical description of these curves is complicated!

Figure: Uniform non-rational B-splines in the plane

Uniform non-rational B-splines: …. one example of a spline curve.

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L5- Curves in GIS

Curves in Geography Markup Language (GML)

GML: defined by Open Geospatial Consortium (OGC) standard is a XML-based grammar for the representation of geospatial data.

A GML Schema allows users and developers to describe geographic

data sets that contain points, lines and polygons and exchange them between

themselves.

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L5- Curves in GIS

Curves supported by GML:

• Arc • Circle • ArcByCenterPoint • CubicSpline • B-Spline • Bezier

Figure: An arc element specified by three points.

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