Not exactly the translation and rotation I mean

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Joella Lewis
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reference and then later translate it to a geodetic frame.

Luckily there are ways we can transform our data through

1 Coordinate Geometry: (). Sometimes it is difficult to work directly in a geodetic based reference coordinate system. In these cases we might start our work in a local frame of reference and then later translate it to a geodetic frame. However, we certainly would not wish to re-do all our work. Luckily there are ways we can transform our data through scaling, translation and rotation. Not exactly the translation and rotation I mean Lect 14 - Oct 19/07 Slide 1 of 7

Coordinate Transformation: (Section 11.8) Conformal coordination transformation is a class (subset) of other linear transformations known as affine transformations.

2 Coordinate Transformation: (Section 11.8) Conformal coordination transformation is a class (subset) of other linear transformations known as affine transformations. Conformal transforms preserve angles so that other affine transforms such as shear and reflection are not conformal. We require two points (A & B) with known coordinates in both the assumed (X-Y) and geo-referenced coordinate system (E-N) and from this we can determine the transformation equations to rotate (), shift (T x,t y ) and scale (s) all other points. = = Az AB(X Y ) Az AB(EN ) = tan 1 X B X A Y B Y A + C X Y = tan 1 E B E A + C EN N B N A s = ( ) EN = ( ) X Y ( E B E A ) + ( N B N A ) for instance ft m ( X B X A ) + ( Y B Y A ) Lect 14 - Oct 19/07 Slide of 7

First we rotate from X - Y to X'-Y' cos sy A sin

cos sy sin + T X N = sx sin + sy cos + T Y

3 First we rotate from X - Y to X'-Y' cos sy A sin sin + sy A cos then we translate (shift) by T X, T Y E A = +T X N A = +T Y these can be combined subing the rotation into the translation E = sx cos sy sin + T X N = sx sin + sy cos + T Y Non-linear Transform Examples Lect 14 - Oct 19/07 Slide 3 of 7

4 Question 11.1 X = X B X A Y = Y B Y A X = Y = X =159.5 ft Y = ft ( ) X Y = ( X B X A ) + ( Y B Y A ) = ( ) X Y = ft A B C E = E B E A N = N B N A E = N = E = m N = m ( ) EN = ( E B E A ) + ( N B N A ) = ( ) EN = m D s = ( ) EN = ( ) X Y a) s = Y = tan 1 X B X A + C X Y = tan 1 E B E A + C EN Y B Y A N B N A = tan = tan = = 68 o 37'44.8" = = 45 o 4'33.6" X = = 68 o 37'44.8"45 o 4'33.6" b) = 3 o 13'11." Lect 14 - Oct 19/07 Slide 4 of 7

5 Find the rotated coordinates of Point A cos sy A sin = cos sin = sin + sy A cos ( ) = sin cos = ( ) A B C c) Find the translation terms T X, T Y from the Point A calc T X = E A T X = = D T Y = N A T Y = = Y d) Find Point C [C X-Y (7041.,6037.3)] E = sx cos sy sin + T X E = ( 7041.cos sin) E = N = sx sin + sy cos + T Y E = ( 7041.sin cos) E = X C EN ( , ) Lect 14 - Oct 19/07 Slide 5 of 7

Coordinate geometry provides another means to accomplish area calculations of polygon shapes very easily.

6 Area Calculations: (Chapter 1). There are a number of methods of determining area in surveying. You ve already determined area from straight line offsets in the field school. Coordinate geometry provides another means to accomplish area calculations of polygon shapes very easily. In additon you can calculate the area of any triangle by: a b area = ss ( a) ( s b) ( s b) where : s = 1 ( a + b + c ) c Lect 14 - Oct 19/07 Slide 6 of 7

subtracting the most easterly boundary from the most westerly.

7 Area by Coordinates: (Section 1.5). We can use coordinates to find the area of any polygon by dividing up the polygon at the intersection points and using the X-axis or Y-axis to determine the individual triangles and trapezoids subtracting the most easterly boundary from the most westerly. area ABCDEA = E EDD E + D DCC D - AE EA - CC B BC - ABB A Expanding with formula we get: area ABCDEA = X E + X D X E + X A ( Y E Y A ) X + X A B ( Y E Y D )+ X + X D C ( Y D Y C ) ( Y A Y B ) X + X B C This expands to : (area) =+X A Y B + X B Y C + X C Y D + X D Y E + X E Y A Y A X B Y B X C Y C X D Y D X E Y E X A ( Y C Y C ) X A X B X C X D Y A Y B Y C Y D Simply list all the x and y ordinates then a) multiply each adjacent X*Y and sum b) multiply each adjacent Y*X and sum c) Find: Area = (Y - X) / X E Y E X A Y A Lect 14 - Oct 19/07 Slide 7 of 7

3D Mathematics. Co-ordinate systems, 3D primitives and affine transformations

3D Mathematics. Co-ordinate systems, 3D primitives and affine transformations 3D Mathematics Co-ordinate systems, 3D primitives and affine transformations Coordinate Systems 2 3 Primitive Types and Topologies Primitives Primitive Types and Topologies 4 A primitive is the most basic