Coordinate Geometry

1
Coordinate Geometry

• Advantages of plane rectangular coordinate system
• Computation examples

2
Common Equations

• Sine rule

3
Common Equations

• Cosine rule

4
Equation for a Line

• Consider two points on a line (XA, YA) and (XB,
  YB)

(XB, YB)
YB - YA
(XA, YA)
XB - XA

• Equation describing line is

5
Length and Direction

• Recall

6
Equation for a Line

• General equation for a line aX bY c 0
• slope -a/b y-intercept -c/b x-intercept
  -c/a

7
Normal form of Equation for Line

• General equation for a line
• aX bY c 0
• Normal form for equation for a line

• Application of normal form

8
Normal form of Equation for Line
aX bY c 0
N
(XP, YP)
9
Translation of Origin
X'p Xp Cx
Y'
Y
Y'p Yp CY
(X'P,Y'P)
(XP,YP)
X'
X
10
Line Intersections
B
D
A
C
11
Evaluating Intersections

• Translate origin to make coordinates smaller
• Subtract 642,990 from X
• Subtract 4,345,290 from Y

12
Evaluating Intersections cont.

• Form equations for lines containing A'B' and C'D'

• Rearrange to general form
• 134X 109Y 11913 0
• 129X 160Y 22952 0

13
Evaluating Intersections cont.

• Solve simultaneous equations

• (1) (2) ? X 124.2
• Substitute into (1) and solve for Y
• ? Y 43.3

14
Evaluating Intersections cont.

• Based on simultaneous equations

• Verify

15
Readings

• Chapter 11 sections 11.1 11.4

16
Area Calculation

• Application
• Issue
• Unit conversion

17
Area by Division in Figures

• Approach

Where s ½ (a b c)
area ½ ab sinC
18
Area by Offsets

• Approach

19
Area by Offsets

• Approach

20
Area by Coordinates

• Approach

Area ½ XA(YD-YB) XB(YA-YC) XC(YB-YD)
XD(YC-YA)
D (XD, YD)
C (XC, YC)
A (XA, YA)
B (XB, YB)
21
Area by Coordinates
D
C
A
B
22
Area by Coordinates
D
C
A
B
23
Area by Coordinates
D
C
A
B
24
Area by Coordinates
area area3 area4 - area1 - area2 ½
(XD-XA) (XC-XA)(YD-YC) ½ (XC-XA)
(XB-XA)(YC-YB) - ½ (XD-XA)(YD-YA) - ½
(XB-XA)(YA-YB) ½ XAYB XBYC XCYD XDYA -
XBYA XCYB XDYC XAYD ½ XA(YD-YB)
XB(YA-YC) XC(YB-YD) XD(YC-YA)
XA YA XB YB XC YC XD YD XA YA
Area ½ sum (plus products) sum (minus
products)
25
Example Calculation by Coordinates
D (XD, YD)
C (XC, YC)
A (XA, YA)
B (XB, YB)
26
Example Calculation by Coordinates
XA YA XB YB XC YC XD YD XA YA
area ½ XAYB XBYC XCYD XDYA - XBYA
XCYB XDYC XAYD ½ 0 443.361 217.036
581.440 151.810 1914.370 1698.096
0 ½ -2522.439 m2 1261.219 m2
27
Area from Maps

• Approaches
• Accuracy of calculated area

28
Scaling Geometric Components

• Example
• Map with scale 11000
• Rectangular area on map dimensions 1.25 in x
  2.35 in
• Solution
• Scale dimensions to ground units

• Calculate ground area 104.17 ft x 195.83 ft
  20,399 ft2

29
Scaling Geometric Components

• Example
• Map with scale 11000
• Rectangular area on map dimensions 1.25 in x
  2.35 in
• Solution
• Calculate area in map units 1.25 in x 2.35 in
  2.9375 in2
• Scale map area to ground area

30
Errors and Mistakes in Area Computation

• Errors
• Mistakes

31
Readings

• Chapter 12 sections 12.1 12.5, 12.9 12.11