Title: Coordinate Geometry
1Coordinate Geometry
- Advantages of plane rectangular coordinate system
- Computation examples
2Common Equations
3Common Equations
4Equation 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
5Length and Direction
6Equation for a Line
- General equation for a line aX bY c 0
- slope -a/b y-intercept -c/b x-intercept
-c/a
7Normal 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
8Normal form of Equation for Line
aX bY c 0
N
(XP, YP)
9Translation of Origin
X'p Xp Cx
Y'
Y
Y'p Yp CY
(X'P,Y'P)
(XP,YP)
X'
X
10Line Intersections
B
D
A
C
11Evaluating Intersections
- Translate origin to make coordinates smaller
- Subtract 642,990 from X
- Subtract 4,345,290 from Y
12Evaluating Intersections cont.
- Form equations for lines containing A'B' and C'D'
- Rearrange to general form
- 134X 109Y 11913 0
- 129X 160Y 22952 0
13Evaluating Intersections cont.
- Solve simultaneous equations
- (1) (2) ? X 124.2
- Substitute into (1) and solve for Y
- ? Y 43.3
14Evaluating Intersections cont.
- Based on simultaneous equations
15Readings
- Chapter 11 sections 11.1 11.4
16Area Calculation
- Application
- Issue
- Unit conversion
17Area by Division in Figures
Where s ½ (a b c)
area ½ ab sinC
18Area by Offsets
19Area by Offsets
20Area by Coordinates
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)
21Area by Coordinates
D
C
A
B
22Area by Coordinates
D
C
A
B
23Area by Coordinates
D
C
A
B
24Area 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)
25Example Calculation by Coordinates
D (XD, YD)
C (XC, YC)
A (XA, YA)
B (XB, YB)
26Example 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
27Area from Maps
- Approaches
- Accuracy of calculated area
28Scaling 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
29Scaling 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
30Errors and Mistakes in Area Computation
31Readings
- Chapter 12 sections 12.1 12.5, 12.9 12.11