Contours: Triangulation

Triangulation is the process of connecting points to form a very large triangular network. Delaunay triangulation is used as a tool using which surfaces can be divided into regions with certain common characteristics. An example of triangular network is shown below

There are numerous applications of triangulation. Triangulation decides the shape of road network for a particular area, decides the pattern of telephone network etc.

The rule guarding the Delaunay triangulation is the circumcircle property, which states that three points are connected to form a triangle if the circumcircle drawn for three points does not enclose any other point within it. To understand this better, look at the points and the circles shown below

From the above logic, it can be learnt that the red circles indicate false triangulation while the black circles show the correct triangulation for the given set of points.

This type of triangulation ensures minimum error as well as reliable data plotting and makes image regeneration possible. Also this method ensures that skinny triangles generating huge circles are also eliminated, thus reducing the amount of error. To understand skinny triangle consider the figure shown below

As seen in the above figure, the arc shown in green generates a really huge circle with respect to the other nearby circles. The cause for such a huge circle generation is that the 3 points considered for the circle have a high angle and produce an obtuse angled triangle known as a Skinny triangle. To overcome this problem a maximum radius is set for the circle arising from the points known as the scanning radius.

The concept of triangulation can also be extended to 3D to generate objects with their true surface features.

Contours/ Interpolate : Point On a 3D Surface

To calculate a point on a 3D plane, assume that there are 3 non-collinear points Pa, Pb and Pc.
Let Pa=[xa,ya,za], Pb=[xb,yb,zb], Pc=[xc,yc,zc].

The general equation of the plane is of the form
Ax+By+Cz+D=0 ………………… (1)

Consider the figure shown below to calculate the coefficients of the above equation A, B and C

Note: X refers to cross product
. refers to dot product
* refers to multiplication

The normal to a plane is of the form
N = (Pac X Pab)/(|Pac X Pab|)
N = (Pc-Pa) X (Pb-Pa)/(| (Pc-Pa) X (Pb-Pa)|)

We find that the resulting direction vectors of the normal from the three points Pa, Pb, Pc are
N = [A,B,C] (The coefficients of the equation of the plane in Equation (1))

Now consider an arbitrary point P=[x,y,z] as shown below

It is found that the plane equation satisfies the following property
(P-Pa).N=0

=>A(x-xa)+B(y-ya)+C(z-za)=0
=>Ax+By+Cz+D=0 (where D=-[Axa+Bya+Cza])
=>x=-(1/A).[By+Cz+D]
=>y=-(1/B).[Cz+Ax+D]
=>z=-(1/C).[Ax+By+D]
Thus depending on whether x, y or z is unknown the equation of the plane can be found.

To understand this consider the example where Pa=[1,2,1], Pb=[1,0,2], Pc=[2,2,1] and P=[4,5,z] where z has to found as shown below

To find z follow the below mentioned steps

Step1: Find the coefficient terms A, B, C, D
Pac=Pc-Pa=(2-1,2-2,1-1)=(1,0,0)
Pab=Pb-Pa=(1-1,0-2,2-1)=(0,-2,1)

The normal to a plane is of the form
N = (Pac X Pab)/(|Pac X Pab|)
N = (Pc-Pa) X (Pb-Pa)/(|(Pc-Pa) X (Pb-Pa)|)

Pac = (Pc-Pa) = [2,2,1] - [1,2,1] = [1,0,0]
Pab = (Pb-Pa) = [1,0,2] - [1,2,1] = [0,-2,1]

(Pac X Pab)= i j k
1 0 0
0 –2 1
= i(0) - j(1) + k(-2)
(|(Pc-Pa) X (Pb-Pa)|)= ?[(0)2 + (1)2+ (2)2]
=√[1+4]
=√5

Substituting in N = (Pc-Pa) X (Pb-Pa)/(|(Pc-Pa) X (Pb-Pa)|)
N=[0 –1 –2]/√5
Therefore A=0 , B=-1/√5 , C=–2/√5

Step2: Find D
D=-[Axa+Bya+Cza]
D=-[0*1 + (-1/√5)*2 + (–2/√5)*1]
=-[0-2/√5-2/√5]
=4/√5

Step3: Find z by substituting the value of x=4 and y=5 in point P
z=-(1/C).[Ax+By+D]
z=-(1/–2/√5).[0*x+(-1/√5*y)+4/√5]
z=(√5/2)*[0+(-1/√5)*5+4/√5]
z=(√5/2)*[0+(-5/√5)+ 4/√5]
z=(√5/2)*[-1/√5]
z=-1/2

Earth Work: Nett Area calculation

Consider the Cross section shown below

Cutting and Filling area needs to be calculated for the portion shown below

The first step towards the calculation involves calculating the distance and elevation at each intersection. Next step is to get both the Cutting area and the filling area, difference of the areas for both the layers is to be calculated at each distance. For example the Cutting area between the distances -7.5 to -6 is calculated by finding the areas for each layer and finding the difference between them. This is as shown below

This Cutting area (A1 – A2) is calculated using the formula

Cutting or Filling Area = ½(Distance 1 + Distance 2) * (Level at Distance 1 - Level at Distance 2)

Similarly area for the entire cross section can be calculated, which is as shown below

Cutting or Filling Area = A1 – A2

The detailed working using Nett area calculation is as shown below

Chainage	6000				Net Filling	7.72						Net Cutting	0.15
Initial Filling Area							Final Filling Area
Distance 1	Distance 2	Level 1	Level 2	Average	Distance	Area	Distance 1	Distance 2	Level 1	Level 2	Average	Distance	Area
-12	-10	77.21	77.21	77.21	2	154.42	-12	-10	77.34	77.34	77.34	2	154.69
-10	-9	77.21	77.34	77.28	1	77.28	-10	-9	77.34	77.34	77.34	1	77.34
-9	-7.5	77.34	77.81	77.58	1.5	116.36	-9	-6	77.34	78.84	78.09	3	234.28
-7.5	-6	77.81	78.36	78.09	1.5	117.13	-6	-5	78.84	78.87	78.86	1	78.86
-6	-4.5	78.36	78.65	78.51	1.5	117.76	-5	-3.5	78.87	78.91	78.89	1.5	118.34
-4.5	-2.25	78.65	78.64	78.65	2.25	176.96	-3.5	-0.2	78.91	79	78.95	3.3	260.55
-2.25	-1.6	78.64	78.67	78.66	0.65	51.13	0.93	3.5	78.98	78.91	78.94	2.57	202.89
-1.6	-1.5	78.67	78.68	78.68	0.1	7.87	3.5	5	78.91	78.87	78.89	1.5	118.34
-1.5	-0.75	78.68	78.71	78.69	0.75	59.02	5	6	78.87	78.84	78.86	1	78.86
-0.75	-0.2	78.71	79	78.85	0.55	43.37	6	7.46	78.84	78.11	78.48	1.46	114.89
0.93	1.1	78.98	78.76	78.87	0.17	13.41	7.46	12	78.11	78.11	78.11	4.54	354.32
1.1	1.5	78.76	78.76	78.76	0.4	31.51
1.5	2.25	78.76	78.74	78.75	0.75	59.06
2.25	3.8	78.74	78.8	78.77	1.55	122.09
3.8	3.85	78.8	78.65	78.73	0.05	3.94
3.85	4.5	78.65	78.67	78.66	0.65	51.13
4.5	6	78.67	78.64	78.66	1.5	117.98
6	7.46	78.64	78.11	78.38	1.46	114.75
7.46	7.5	78.11	78.1	78.11	0.04	2.81
7.5	9	78.1	77.13	77.62	1.5	116.43
9	10	77.13	76.9	77.02	1	77.02
10	12	76.9	77.31	77.11	2	154.22
					Total	1785.63						Total	1793.35
Initial Filling Area							Final Filling Area
Distance 1	Distance 2	Level 1	Level 2	Average	Distance	Area	Distance 1	Distance 2	Level 1	Level 2	Average	Distance	Area
-0.2	0	79	79.1	79.05	0.2	15.81	-0.2	0	79	79	79	0.2	15.8
0	0.75	79.1	79.2	79.15	0.75	59.36	0	0.93	79	78.98	78.99	0.93	73.46
0.75	0.93	79.2	78.98	79.09	0.18	14.24
					Total	89.41						Total	89.26

Earth Work: Trapezoidal method

The trapezoidal method is a used to get the area. This is done by inscribing or circumscribing n number of trapezoids and triangles. The areas of the trapezoids and triangles are then summed to get the total area. Before going to the Trapezoidal method in detail first lets us see how the area calculation in done for triangle and trapezoid.

Trapezoidal perimeter
P = b1 + b2 + c + d
= 10 + 6 + 10.2 + 10.2
= 36.4

Trapezoidal Area
A = 1/2 * a * (b1+b2)
= 1/2 * 10 * (10+6)
= 80

Triangle

Triangle perimeter
P = b + c + d
= 10 + 12.81 + 10.2
= 33.01

Triangle area
A = a * b/2
= 10 * 10 /2
= 50

To understand the Trapezoidal method better consider a portion of the cross section shown below.

Here calculations for cutting and filling Area for segments I, II and III are shown below.

I) Segment I between –10 & -5 is a triangle
Formula for triangle = ½ * Breadth * Height
Height = 0.1 (cutting)
Breadth = 5
So Cutting Area = ½ * 0.1 * 5 = 0.25

II) Segment II between –5 & -2 is made of two triangle (as both cutting and filling is there)

Formula for triangle = ½ * Breadth * Height

First find the intersection point:
Depth of cutting = 0.10
Depth of filling = 0.20
Width = 3

Intersection point = -5 + 0.10 / (0.10 + 0.20) * 3 = -4
Width for cutting = 1
Width for filling = 2

First Triangle (Cutting) - 2
Width = 1
Depth = 0.1
So Filling Area = 1 * 0.1 / 2 = 0.05

Second Triangle (Filling) - 3
Width = 2
Depth = 0.2
So Cutting Area = 2 * 0.2 / 2 = 0.2

III) Segment III between –2 & 0 is made of trapezoid

Area of trapezoid = ½ * a * (b1 + b2) / 2
Area calculation for trapezoid = ½ * Width * ½ (Height 1st Line + Height of 2nd Line)

Area of trapezoid (Filling) – 4
Height 1 = 0.2
Height 2 = 0.5
Width = 2

So Filling Area = ½ * 2 * (0.2 + 0.5) = 0.7

Earth Work: Triangulation Method

Triangulation is the process of connecting points to form a very large triangular network. Delaunay triangulation is used as a tool using which surfaces can be divided into regions with certain common characteristics. An example of triangular network is shown below

There are numerous applications of triangulation. Triangulation decides the shape of road network for a particular area, decides the pattern of telephone network etc.

The rule guarding the Delaunay triangulation is the circumcircle property, which states that three points are connected to form a triangle if the circumcircle drawn for three points does not enclose any other point within it. To understand this better, look at the points and the circles shown below

From the above logic, it can be learnt that the red circles indicate false triangulation while the black circles show the correct triangulation for the given set of points.

This type of triangulation ensures minimum error as well as reliable data plotting and makes image regeneration possible. Also this method ensures that skinny triangles generating huge circles are also eliminated, thus reducing the amount of error. To understand skinny triangle consider the figure shown below

As seen in the above figure, the arc shown in green generates a really huge circle with respect to the other nearby circles. The cause for such a huge circle generation is that the 3 points considered for the circle have a high angle and produce an obtuse angled triangle known as a Skinny triangle. To overcome this problem a maximum radius is set for the circle arising from the points known as the scanning radius.

The concept of triangulation can also be extended to 3D to generate objects with their true surface features.

Interpolation: Polyline and 3D Polyline

Polyline

• Polyline is a connected sequence of line segments created as a single object.
• It is a 2D object i.e. it has only x and y co-ordinates.
• It is possible to create straight-line segments, arc segments, or a combination of the two.
• They are ideal for applications like Contour lines for topographic, scientific and other applications

3D Polylines

3D polylines are same as polylines but resulting in a 3D-polyline-object type. These 3D polylines contains x, y and z coordinate values.

Interpolation: Straight line method

One of the ways to do interpolation is Straight line method. This method is used when the values are to be interpolated using only two levels. Following figure explains the straight line method

Here the values of P and Q are given and the distance between P and Q is also given. Now say value at point R is needed which is at a distance of 1 from P. Using straight line method the value at R is calculated as shown

R = P + [(Q - P) * Distance between P and R]
(Distance between P and Q)
R = 10 + [(20 - 10) / 4 * 1]
R = 10 + [-2.5]
R = 12.5

Weighted Average Method

Weighted Average Method is one of the ways to do interpolation. This can be used when a single points value is needed which is not in straight line with the two points. Following figure explains the weighted average method

Here given are the points P, Q and R where P and Q have values and the value at point R is required to be calculated which is at a distance of 2 from point P and 1 form point Q. Using Weighted average method R value is calculated as shown

R = [(P * Distance from P to R) + (Q * Distance from Q to R)]
(Distance from P to R + Distance from Q to R)

R = [(10 * 2) + (20 * 1)] / (2 + 1)
R = 13.333

TopoDraw

In the field of civil engineering, while doing survey there are innumerable objects found. All these Objects are termed as Survey Blocks. These survey blocks are later on joined while plotting it on various CAD software for further processing. Every survey block is assigned a linetype so that identification of these survey blocks becomes easy while processing. Few of the survey blocks are listed below

Sl.no	Survey Blocks	Code	Linetype	Symbols /Block
1	GPS point	GP -1(no)		GP
2	Total Station Point	ST -2(no)		ST
3	GTS Bench Mark	GTS – (value)		GTS
4	Bench Mark	BM –(1no)		BM
5	Survey Stone	SVY -ST		SVY
6	Village Boundary Stone	VB -ST		VB -SVY
7	Optical Fibe stone	OFC		OFS
8	Kilometer Stone	KM		KM
9	Parlong Stone	P-ST		P-ST
10	Highway/Road Boundary Stone	HRB/RB		HRB/RB
11	Railway Boundary Stone	RBS		RBS
12	Property Stone	PB-ST		PB
13	Left ASP Edge	LASP-01		TP
14	Right ASP Edge	RASP		EP
15	Left Road Edge	LRD		LP
16	Right Road Edge	RRD
17	Left Mud Road Edge	LMRD
18	Right Mud Road Edge	RMRD
19	Foot Path	FP
20	Pavement Foot Path Top	PFPT
21	Pavement Foot Path Bottom	PFPB
22	Left Median Top	LMT
23	Right Median Bottom	LMB
24	Telephone Pole	TP
25	Electrical Pole	EP
26	Lamp Pole	LP
27	Hi-Tension Pole(more then 33 kV)	HTP
28	PYLON	PLY
29	Electrical Box	EBOX		EBOX
30	Transformers	TF		TF
31	Junction Box	JB		Junction Box
32	Post Box	PB		PB
33	Dust Bin	DB		DB
34	Tap	TAP		TAP
35	Hand Pump	HP		HP
36	Bore Well	BW		BW
37	Well	Well		Well
38	Flag Pole	FP		FP
39	Man Hole	MH		MH
40	Valves	Valve		Valves
41	Left Lind Drain Top	LLDT
42	Right Lind Drain Bottom	RLDB
43	Right Lined Drain Top	RLDT
44	Drain Center	DC
45	Left Lined Drain Bottom	LLDB
46	Left Unlined Drain Top	LULDT
47	Right Unlined Drain Bottom	RUDB
48	Chambers	CH		CH
49	Gate	GAT		Gate
50	Barricade	BAR		Barricade
51	Solder Edge	SEG
52	Embakement Top	Ecut –T
53	Embakement Bottom	Ecut –B
54	Cutting Top	Cut –T
55	Cutting Bottom	Cut -B
56	Culvert	CUL
57	Building 1	BDG-1
58	Building 2	BDG-2
59	Building 3	BDG-3
60	Temporary Shop	HUT/SH
61	Nala Top	Nal-T
62	Nala Bottom	Nal-B
63	Bridge	BRDG
64	Compound Wall	CW
65	Fence	FE
66	Water Tank	TNK
67	School	BDG/SC
68	Public Building	BDG
69	Retaining Wall	RW
70	Temple	TM
71	Church	CH
72	Mosque	MQ
73	Grave	GY
74	Over Head Tank	OHT
75	Tiled House	BDG/TH
76	Traffic Signal Board	TSB
77	Pond Top	POT
78	Pond Bottom	POB
79	Name Board	NB
80	Pump House	PH
81	Jungle Tree	T(J)
82	Baniyan Tree	T(B)
83	Neem Tree	T(N)
84	Coconut Tree	CT
85	Babool Tree	T(B)
86	Bamboos	T(Bab)
87	Plantation	T(P)
88	Line of tree	T(Line)
89	River Top	Riv-T
90	River Bottom	Riv-B
91	Road Median /Island	RM(RI)
92	Wing Hall	RW
93	Weir	VR
94	Pit	PIT
95	Pit Center	PIT-C
96	Guard Stone	G-ST
97	Field Bund	FB
98	Pier	PIE
99	Rocky Area	RCK
100	Canal Top	CAN-T
101	Canal Center	CAN-C
102	Petrol Bunk	P-Bunk
103	HUT	HUT
104	Weight Bridge	W-BDG
105	Water Level	WL
106	Pipe Line Edge	PLE
107	Pipe Line Top	PLT

Some Commonly used AutoCAD Terms

Layers

As the Drawings becomes more and more complicated, organization of the drawing objects becomes complex too. To manage these objects layers can be used. Layers can be defined as a set of transparent overlays as shown below

Using layers is similar to using overlays in a manual drafting environment where clear media that contain groups of related design elements are placed throughout the overall design. Each layer has property settings that determine the colour, line type, and line weight of the objects. The end result is that when all the layers are combined then it can be seen as one complete drawing, and has the flexibility to easily remove the overlays to focus only on certain aspects of the design. See the example below where the final drawing looks like the one shown below but actually it is achieved when objects in all the layers are seen together

The above is got when all the following layers are seen together

Here in this layer only the road profile is drafted.

Here in this layer only the dimensions are drafted.

To understand the concept of slope and gradient consider the figure shown below

The above figure represents Gradient where the person walks horizontally 1m for every 1m height this is also told as 1 in 1 and is represented as 1:1.

The above figure represents Slope where the person walks 100m horizontally for every 1m height this is also told as 1 in 100.

Datum

Datum is the point which is obtained with reference to the combination of the vertical scale and horizontal scale when joined at a common point to form a reference coordinate system with reference to which a given quantity is measured on a system. See the figure shown below

To understand as to why datum is to be used, consider the example shown below where the lines start at the value 54.

It can be seen that below the object, lot of space is wasted and hence reducing the readability of the graph. Now if the graph starts at 50 rather than starting from 0 the readability is increased. This value 50 is called datum. This is as shown below