engleski

New Method for Reducing Sharp Corners in Cartographic Lines with Area Preservation Property

In this paper an area preservation function for modification of polylines is presented. For given three consecutive points Ti, Ti+1 and Ti+2 in polyline the function returns four consecutive points Ti, Q, S and Ti+2. There are four unknowns: xq, yq, xs and ys, therefore four independent constraints are necessary. The first is area preservation, i.e., the area of the triangle Ti, Ti+1, Ti+2 is equal to the area of the quadrilateral Ti, Q, S, Ti+2. Other three constraints are chosen to ensure simplicity and applicability. To ensure that new segments are not too long or too short the lengths of new segments are chosen to be equal. We use fourth constraint to define the angles among new segments. We define that smaller angle of two in points Q and S gets its maximum possible value. This will be true when the angles in points Q and S are the same. In this way, Ti, Q, S, Ti+2 form an isosceles trapezoid. To find its elements and the coordinates of the points Q and S the fourth order polynomial has to be solved. We prove that there is always one and only one solution to the problem. The solution is given in closed form using Ferrari’s method. Using that function, we should be able to reduce sharp corners in polylines which result from the generalization process. The result of such an application is presented.

area preservation; polyline; cartographic generalization

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