1. 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

2. 2 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)

3. 3 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!

4. 4 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 function y=f(x) Dashed: a general curve in the plane

5. 5 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. A polynomial of degree n is written as:

6. 6 L5- Curves in GIS Example: create a 3 rd order polynomial that passes through the 4 points in the table. Linear system of equations: 3 rd order polynomial

7. 7 L5- Curves in GIS The fitted polynomial of 3 rd degree to the points The points

8. 8 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

9. 9 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

10. 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. 1 st polynomial 2 nd polynomial 3 rd polynomial 10

11. 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 : 11

12. L5- Curves in GIS Hermite interpolation … based on interpolation with piecewise polynomials (3 rd degree). Requirements: 1)the interpolated curve must pass through all points (each 2 following points) 2)the first order derivative must be continuous EVERYWHERE. 12

13. L5- Curves in GIS Figure: parameters in Hermite interpolation 13

14. 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) 14

15. L5- Curves in GIS Requirements: 2)the first order derivative must be continuous everywhere. This function is continuous within a polynomial. Question: but what happens between the polynomials? the derivatives at the start and end points: 15

16. 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. 16 So all the constraints are satisfied and, hence, the equation provides the Hermite interpolation.

17. 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). solid dashed 17

18. 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. 18 A standard method is to use the following differences:

19. L5- Curves in GIS Cubic spline A piecewise 3 rd 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 parameters passing 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 19

20. 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. 20

21. L5- Curves in GIS How to compute natural cubic splines? 21

22. L5- Curves in GIS 2) apply the Hermite interpolation function 22

23. 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 23 Figure: Natural cubic spline interpolation of the points. The first order derivatives at the points are given in the table above.

24. L5- Curves in GIS Curves in the plane This 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

25. L5- Curves in GIS General expressions for the curve the curves in the plane are represented using cubic polynomials Matrix form: 25

26. 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 points Ri are the tangent vectors of the same curve 26

27. 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 27

28. 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

29. L5- Curves in GIS Spline curves in the plane The 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. 29

30. 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. 30

31. L5- Curves in GIS Curves supported by GML: Arc Circle ArcByCenterPoint CubicSpline B-Spline Bezier Figure: An arc element specified by three points. 31