MCP Manual - Exercise 5 Polar and Cartesian Coordinate Systems

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Exercise 5 Polar and Cartesian Coordinate Systems

MapScenes Systems CTS DO NOT DUPLICATE | 2008 MicroSurvey Software Inc. | 1

Exercise # 5 Polar and Cartesian Coordinate SystemsObjective:

After this exercise, you will be able to;

1. Understand how Baseline Offset Measurements and Theodolite Measurements areused in a forensic map

2. Convert coordinates from Polar to Cartesian.

3. Compute elevations from scene measurements.

4. Describe how MapScenes uses trigonometry to create a map in an Assumed,Cartesian Coordinate system.

START

Scientific Calculator

You will need to use a scientific calculator for this exercise:

Open Start | Programs | Accessories | Calculator

Open View and set the options as shown:

If your version of Windows does not have the calculator, install the MicroSurvey Calculatorfrom your student disk or from http://www.microsurvey.com/calc/index.htm.

Now we are ready to begin the calculator exercise.


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2 | MapScenes Systems CTS DO NOT DUPLICATE | 2008 MicroSurvey Software Inc.

Polar and Cartesian Coordinate Systems

Baseline Offset Method Review:

Recall that the traditional Baseline Offset method for creating a forensic map relies on a

distance down a Baseline and a distance at right angles to the left or right of that line tolocate evidence:

This method of positioning evidence is referred to as a Cartesian coordinate system, inwhich a unique position is defined by a Y coordinate (the baseline distance) and an Xcoordinate (the offset distance.)


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Polar Coordinates:

Evidence Recorder and MapScenes allow forensic mappers to use a Theodolite and EDM tomeasure a direction and distance to any evidence to be mapped. The Theodolite is locatedat either end of the baseline and will measure an angle clockwise from the baseline line.

In the example shown, evidence is measured by aTheodolite set up at Reference Point 1 at the Southerlyend of the baseline. The Northerly end of the Baseline isReference Point 99. In forensic mapping, reference pointsare often marked by a concrete nail or other permanentmarker.

The evidence at the small square is located by measuring

the HCR, an angle of 33 degrees to the right of the baseline,and a distance from the Theodolite of 10.51 feet.

A positive angle is always measured clockwise, or left toright.

This method of positioning evidence is referred to as aPolar coordinate system, in which a unique position isdefined by a direction and distance.

Computing Cartesian Coordinates from Polar Coordinates

A forensic mapper will locate every piece of evidence by measuring and recording an HCR(Horizontal Circle Reading,) VCR (Vertical Circle Reading) and (SD) Slope Distance. In orderto display the position of each piece of evidence, MapScenes needs to calculate a Cartesiancoordinate for each piece of evidence.

It is valuable to understand how Cartesian Coordinates are calculated from HCR, VCR andSlope Distance. Mapscenes and Evidence Recorder use Trigonometry to calculate the

Cartesian Coordinates. This is made easy to understand if you use the SOHCAHTOA ruledescribed below:

SOHCAHTOA

A way of remembering how to compute the sine, cosine, and tangentof an angle.


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SOH stands for Sine equals Opposite over Hypotenuse.

CAH stands for Cosine equals Adjacentover Hypotenuse.

TOA stands for Tangent equals Opposite over Adjacent.

Applying SOHCAHTOA to Compute Cartesian Coordinates

Opposite Side = X of evidence

Adjacent Side = Y of evidence

Hypotenuse = Measured Distance to Evidence

= Horizontal Circle Reading, or HCR


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Example:

, or HCR = 33

Hypotenuse, or Distance = 10.51 feet:

The X coordinate is calculated by:

Sin 33 = X /10.51

Sin 33 x 10.51 = X

Try this in the calculator and see if your answer is:

5.72 = X

The Y coordinate is calculated by:

Cos 33 = Y /10.51

Cos 33 x 10.51 = Y

Try this in the calculator and see if your answer is:

8.81 = Y

MapScenes stores each point in the form below, remembering each evidence point with apoint number, an X and a Y. The Z value shown represents an elevation, which will beexplained later.

Calculator Exercise:

Using the formulas in the example above, compute an X and Y coordinate for the examplesthat follow:


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X = Sin 47 x 12.18

X = ___________

Y = Cos 47 x 12.18

Y = ____________


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Introducing the Origin

We have assumed that the X, Y coordinates of RP (1) are 0,0. RP (1) is the Origin of boththe Cartesian and Polar coordinate systems. The X and Y values we have added to them arein fact a difference of X and Y values between the starting point and the end point.

RP(1) does not have to be at 0,0. It can be located by any pair of numbers, and using thedifference in X and Y values between the origin and evidence allows us to compute absolutecoordinates for the evidence. In the example below the Origin is at 100,100:

X,Y (Evidence) =

(X (Origin) + Relative X), (Y (Origin) + Relative Y)

X (Evidence) = 100 + 5.72

Y (Evidence) = 100 + 8.81

X,Y (Evidence) = 105.72, 108.81


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Calculator Exercise:

Compute Absolute Coordinates for the Shoe:

Remember to subtract negative values.

X,Y RP(1) = 250, 350

X, Y (Shoe) = ____, ____


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Compute Absolute Coordinates for the Bumper:

Remember to subtract negative values.

X,Y RP(1) = 325.87, 1025.59

X, Y (Bumper) = _______, ________

Horizontal and Vertical Distance

In the above examples, things have been simplified. We have been using a horizontaldistance to compute relative X and Y values. A Theodolite does not simply measure ahorizontal distance as implied, but measures a Slope Distance and Vertical Circle Reading:


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Trigonometry is used to compute the Horizontal Distance and Vertical Distance:

= (90 VCR)

Opposite = Vertical Distance

Adjacent = Horizontal Distance

Applying SOHCAHTOA:

VD = SIN (90-VCR) x SD

HD = COS (90-VCR) x SD


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Example:


VD = SIN (90 83) x 20.78

VD = SIN 7 x 20.78

VD = 2.53


HD = COS (90 83) x 20.78

HD = COS 7 x 20.78

HD = 20.63


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Calculating Horizontal and Vertical Distances

VD = SIN (90-VCR) x SDVD = ____________


HD = ____________


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Height of Instrument and Height of Target

Vertical Distance from the reference point is recorded in order to compute the elevation, orZ value of evidence. Because Vertical Distance is measured from the horizontal axis of the

theodolite to the center of the prism the height of the theodolite and the height of the target,or prism must be considered.

These values are called:

Height of Instrument (HI)

Height of Target (HT)

To calculate the elevation at RP (2) in the diagram above:

Elev RP(2) = Elev RP(1) + HI + VD HT


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Calculating Elevations Measured with a Prism Pole


Elev RP(2) = Elev RP(1) + HI + VD HT

Compute the elevation at RP(2)

Elev RP (2) = ________________


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Calculating Elevations Measured Using Reflectorless Mode

Some EDMs are able to measure short distances without a prism. In this case the HT is setto zero:

Compute the elevation at the bullet hole.

Elev Bullet Hole = ________________


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MCP Manual - Exercise 5 Polar and Cartesian Coordinate Systems

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