Computational ?P1 + (1- ?) P2. P3 is any

Computational
Geometry

Introduction:

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Computational geometry is branch of
software engineering, examines calculations for taking care of geometric issue.
In present day Mathematics, Computational graphs has likewise utilized as a part of PC computer
graphics, robotics, VLSI design, computer-aided
design, molecular modeling, metallurgy, manufacturing, textile layout,
forestry, and statistics.

Here, we studies few computational-geometry
algorithms in 2-D, representing each input object by a set of points {p1, p2, p3…}, where each pi = (xi, yi  )and xi , yi  € R.
Computational geometry can also apply to 3-D and even higher-dimensional spaces
such as 4-D and 5-D,but such problems and their solutions can be very difficult
to visualize.

Line-Segment
Properties:

A convex
combination of two distinct points p1=(x1, y1) and p2 =(x2, y2) is any
point p3 =(x3, y3) such
that for some ? in the range 0 <=?<=1, we have    x3 = ?x1 + (1- ?) x2    and    y3 = ?y1 + (1- ?) y2. We also write that p3 = ?P1 + (1- ?) P2. P3 is any point that is on the line passing through p1 and p2 and between p1 and p2 on the line. Given two distinct points p1 and p2, the line segment p1p2 is the set of convex combinations of p1 and p2. We call p1 and p2 the endpoints of segment p1p2.There are some properties of line segment. 1.     Given two directed segments p0p1 and p0p2, is p0p1 clockwise from p0p2 with respect to their common endpoint p0? 2.      Given two line segments p0p1 and p1p2, if we traverse p0p1 and then p1p2, do we make a left turn at point p1 ? 3.     Do line segments p1p2 and p3p4 intersect?  Our methods use only additions, subtractions, multiplications, and comparisons. Because they are not computationally expensive. We use neither division nor trigonometric functions, both of which can be computationally expensive and with round-off error.   Picture ( a) shows that The cross product of vectors p1 and p2 is the shaded area of the parallelogram Picture (b) shows that  The lightly shaded region contains vectors that are clockwise from p. and The darkly shaded region contains vectors that are counterclockwise from p.   Cross Product: The cross item is a 3-D idea. According to the "right-hand rule" , a vector that is orthogonal to both p1 and p2 and whose magnitude is . Processing cross items lies at the core of our line-segment methods. Consider vectors p1 and p2 as shown above in Picture(a). We can interpret the cross product p1 p2 as the signed area of the parallelogram formed by the points (0,0), p1 , p2, and p1 + p2 = (x1+x2, y1 + y2) .An equivalent, but more useful, definition gives the cross product as the determinant of a matrix: p1 is clockwise from p2 with respect to the origin (0,0), If p1 p2 is positive. p1 is counterclockwise from p2 if  Cross Product result is negative. And Picture (b) above demonstrates the clockwise and counterclockwise areas in respect to a vector. If the cross item is 0, limit condition emerges; in this case, the two vectors are collinear, either pointing in similar directions or Inverse directions. To determine whether a directed segment p0p1 is closer to a directed segment p0p2 to their common endpoint p0 in a clockwise direction or in a counterclockwise direction, That is, we define p1-p0 denote the vector   and And we define p2-p0 similarly.   In the above example here check that the given line-Segments are clockwise or anti clockwise. if clockwise, make right turn and if anticlockwise make left turn. If cross product is positive then it is clockwise otherwise anticlockwise. Determining whether consecutive segments turn left or right: Here question arise whether it turns right or left. We compute the cross product, if the cross product is negative then it is anticlockwise and make a left turn. And if cross product is positive then clockwise and make a right turn.  If a cross product is 0 it means that points p0, p1 , and p2 are collinear. Determining whether two line segments intersect: If two line-segments intersect, we check whether each segment straddles the line containing the other. A segment p1 p2 straddles a line if point p1 lies on one side of the line and point p2 lies on the other side. A boundary case is if p1 or p2 lies directly on the line. Two line segments intersect if and only if these conditions holds: 1. Each segment straddles the line containing the other. 2. Boundary Case condition: An endpoint of one segment lies on the other segment. There implements an algorithm called SEGMENTS-INTERSECT return TRUE if segments p1 p2 and p3p4 intersect and if they do not return FALSE. It calls the subroutines DIRECTION, which perform relative orientations using the cross-product, and ON-S EGMENT, which determines whether a point is collinear with a segment lies on that segment. Other applications of cross product: Here introduce additional uses for cross products: ·        Sort the set of point with respect to their polar angles. ·        Comprising in sorting. ·        Use red-black trees to maintain the vertical ordering of a set of line-segments. ·        Two segments that intersect a given vertical line

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