Relative Location of a Point with Respect to a Straight Line (0,0) 5 5 (2, 2) (4, 5) (0, 5) (6, 3) -3x + 2y +2 = 0 s = A x t + B y t + C s < 0 s > 0

Perpendicular Distance between a Point and a straight Line (0,0) 5 5 (2, 2) (4, 5) (6, 3) -3x + 2y + 2 = 0 (4, 0) d -3x + 2y + 12 = 0 LL’L’ s = C - C’ =

Midpoint (0,0) 5 5 p (x 1 y 1 ) q (x 2 y 2 ) r (x r y r ) x r y r (x 1 + x 2 ) / 2 (y 1 + y 2 ) / 2 =

Perpendicular Bisector of a Line Segment (0,0) 5 5 p (x 1 y 1 ) q (x 2 y 2 ) P X Y x1 – x2 y1 – y2 ½ [ (x y 2 2 ) – (x y 1 2 ) ] ABCABC =

Intersection of Two Lines (0,0) x +2y +2 = 0 2x + 4y – 28 = 0 (4, 5) X Y Ax + By + C = 0 Ex + Fy + G = 0 (GB – FC) / (FA – EB) (CE – AG) / (FA – EB) xiyixiyi =

Using linear equation form requires a two-step process: first, the line intersection is calculated and then checks are made to determine if this point lies on each line segment Intersection of Two Line Segments (1)

For two segments, S and S’, connecting points (x 1, y 1 ) to (x 2, y 2 ) and (x 1 ’, y 1 ’) to (x 2 ’, y 2 ’) respectively, their intersection is determined by solving for h = D 1 / D and h’ = D 2 / D, where D = (x 2 ’ - x 1 ’) (y 1 - y 2 ) – (x 2 - x 1 ) (y 1 ’ - y 2 ’) D 1 = (x 2 ’ - x 1 ’) (y 1 ’ - y 1 ) – (x 1 ’ - x 1 ) (y 2 ’ - y 1 ’) D 2 = (x 2 - x 1 ) (y 1 ’ - y 1 ) - (x 1 ’ - x 1 ) (y 1 - y 2 ) Intersection of Two Line Segments (2)

Angle Bisector of Two Intersected Lines (0,0) Y X Ax + By + C = 0 Ex + Fy + G = 0 Rx + Sy + T = 0 θ = RSTRST A’ + E’ B’ + F’ C’ + G’

Different Angle Bisectors Depending on the Direction of the Intersecting Lines (0, 0) (a) (b) (c) (d)

Measurement of the Length of a String Length of a String: (x 1 y 1 ) (x 2 y 2 ) (x 11 y 11 )

Measurement of the Area of a Polygon – Using Triangular Decomposition (0,0) Y X

Vector Product (Review) Let a, b, c be vectors, a and b have the same startpoint Let | c | = | a | | b | sinθ The direction of c is perpendicular to the plane decided by a and b, and according to the right hand law c = a x b = (x i+1 i + y i+1 j) x (x i i + y i j) = x i+1 y i – x i y i+1 where i, j, are the basic unit vectors, i x i = j x j = 0, i x j = k, j x i = -k, k = 1 (0, 0)(x i+1 y i+1 ) (x i y i ) θ a b

Measurement of the Area of a Polygon – Using Trapezoidal Decomposition (0,0) Y X

Area Measurement of a Complex Polygon (0,0) Y X

Area Calculation in a Tessellation( 剖分 ) of Space Number of Areal Units = (1/2) b + c -1 (a)(b)

Summary We have focused on the basics of analytical geometry that comprise the primitive tasks within more complex algorithms in digital cartography. However, the emphasis has been on the coordinate properties of maps as absolute locations and not their relative position within a digital image. In the following chapter, the data organization of digital maps is discussed and topological properties are added to the locational ones.

Questions for Review (1) (0,0) Y X p (5, 4) A symbol with the representative point p (5, 4) needs to be rebuilt upside down, reduced in its scale to the half and the point p needs to be placed at a new location (3, 1). Please give the sequence of the matrices for point transformations.

A straight line was built in such a way that it passes through two points p (2, 2) and q (4, 5) sequentially. In which halfplane does another point t (1, 4) lie with respect to this straight line? What is the perpendicular distance between point t and the line pq? We know that one can also build an equation for a perpendicular bisector （中垂线） of a line segment using the midpoint and the negative reciprocal number of the slope of the given line segment. Compared with the method in this class, which one is better? Why? Questions for Review (2)

Questions for Review (3) (2, 5) 5 5 (5, 4) (6, 2) (4, 1) (3, 2) (2, 1) (1, 3) Please cauculate the perimeter and the area of this polygon.

Data Structures The central object in a digital cartographic processing system is the digital representation Real World → Data Model → Data Structure → Hardware Storage Data structures form the basis of software design File structures are representation of the objects in hardware storage

Topology: a branch of geometry, concerned with a particular set of geometrical properties – those that remain invariant under topological transformation Topological Transformation: the transformation induced by stretching on a rubber sheet is called a topological transformation or homeomorphism Basic Topological Concepts (1)

Basic Topological Concepts (2) A topological space is a set of objects, T, together with a collection of open subsets of T, denoted as { t }, called a topology on T, that satisfy the following axioms: 1) the intersection of any finite collection of open subsets is open 2) the union of any collection of open subsets is open

Neighborhood A subset S is a neighborhood of any object, s, that is an element of the set T, if there is a member R that is an element of { t } such that s is an element of R and R is a subset of S. R S T s { t } E

Neighborhood of an e­Ball e -ball if all points of e -ball associated with any point in Hubei Province would also only contain points that are part of Hubei. In Euclidean R n space, a subset S is a special neiborhood of the point p called an e -ball if all points of R n within an epsilon distance ( a very small distance) of p are contained in S. An e -ball associated with any point in Hubei Province would also only contain points that are part of Hubei.

Set S and Its Complement S S’

Closure S - SS-S- The closure S - of a subset S is the intersection of all closed subsets that contain S.

Interior of S S°S° The interior of a subset S, denoted as S°, is the set of all the elements of S for which S is a neighborhood.

Boundary of S jSjS The boundary of a subset S, denoted as jS, is the set of all points in the intersection of the closure of S and the complement S’.

Near Point Let S be a topological space. Then S has a set of neighborhoods associated with it. Let X be a subset of points in S and x an individual point in S. Define x to be near X if every neighborhood of x contains some point of X. Near to C Neighborhood seperating from C Not near to C Open unit circle C

Connected and Disconnected Sets Let S be a topological space and X be a subset of points of S. Then X is connected if whenever it is partitioned into two non-empty disjoint subsets, A and B, then either A contains a point near B or B contains a point near A, or both. ( a )( b )( c )

Cells Connected subsets are also called cells. In a 2-D surface, the primary connected subsets are 0-cells (points), 1-cells (lines) and 2-cells (areas). 0-cell 1-cell2-cell

Topological Incident Relationship A pair of these objects having different dimensions is incident with each other if their intersection is not a null set. N E1E1 E2E2 E3E3 E4E4 A E1E1 E2E2 E3E3 ( a )( b )

Topological Adjacent Relationship Two objects of the same dimension are adjacent to each other if they share a bounding object. E N1N1 N2N2 EA1A1 A2A2 ( a )( b )

Topological Inclusive Relationship If the union of set A and set B equals set A, then set A includes set B. The dimension of set A is equal or greater than set B. A B1B1 B2B2 B3B3

What is topology ? What is a topological transformation? What is a topological space? What is a topological property? What is the concept of neighborhood of a subset S to an object s? What is the definition of a near point x to a subset X? Do the definitions of connected and disconnected sets accord with our intuition? Questions for Review (4)

Between which kinds of objects does the incident relationship exist? Which kind of object bounds which kind of objects? What does the word ‘cobound‘ mean? Between which kinds of objects does the adjacent relationship exist? If set A includes set B, which has the dimension 1, what kinds of dimension can set A be? Questions for Review (5)