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Data Collection and Analysis�Area Measurement

How to measure the perimeter and area of an arbitrary closed shape?

2 of 11

Data Collection and Analysis�Area Measurement

How to measure the perimeter and area of an arbitrary closed shape?

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Consider Descartes’ method:

Could we do this by driving a robot around the shape?

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From Historical Connections in Mathematics, Volume III, Area the Easy Way

INSTRUCTIONS

  1. Beginning with any vertex, list the coordinates of the vertices in order, moving counter-clockwise around the polygon. List the first pair again at the end.
  2. Find the diagonal products from left to right.
  3. Find the diagonal products from right to left.
  4. Sum each column of products.
  5. Find their difference and divide by 2.

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This is the polygon’s area.

3 of 11

Data Collection and Analysis�Area Measurement

Need to generate (x, y) pairs as the robot follows the line around…

​

But unlike the plotted curve, the robot does not know what the next pair will be.

​

Need to find a way to remember the last point and calculate the next one.

​

Just a little trigonometry…

​

and a compass.

​

Inspired by Area the Easy Way

(X old, Y old)

North

Distance

(X new, Y new)

Heading

ΔX

ΔY

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

4 of 11

Data Collection and Analysis�Area Measurement

Need to generate (x, y) pairs as the robot follows the line around…

​

But unlike the plotted curve, the robot does not know what the next pair will be.

​

Need to find a way to remember the last point and calculate the next one.

​

Just a little trigonometry…

​

and a compass.

​

Inspired by Area the Easy Way

(X new, Y new)

Distance

(X old, Y old)

North

Heading

ΔX

ΔY

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

Sine and cosine provide positive and negative factors that scale the sides of the right triangle.

5 of 11

Data Collection and Analysis�Area Measurement

Show development of the path as the robot follows along the orange curve.

​

Inspired by Area the Easy Way

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

Sine and cosine provide positive and negative factors that scale the sides of the right triangle.

6 of 11

Data Collection and Analysis�Area Measurement

Show development of the path as the robot follows along the orange curve.

​

Inspired by Area the Easy Way

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

Sine and cosine provide positive and negative factors that scale the sides of the right triangle.

ΔX

ΔY

7 of 11

Data Collection and Analysis�Area Measurement

Example path showing generated (x, y) pairs as the robot follows the line around.

(X1, Y1)

(X0, Y0)

North

ΔX

ΔY

(X2, Y2)

​

Inspired by Area the Easy Way

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

Sine and cosine provide positive and negative factors that scale the sides of the right triangle.

8 of 11

Data Collection and Analysis�Area Measurement

Example path showing generated (x, y) pairs as the robot follows the line around.

(X1, Y1)

(X0, Y0)

North

ΔX

ΔY

(X2, Y2)

(X3, Y3)

(X4, Y4)

(X5, Y5)

​

Inspired by Area the Easy Way

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

Sine and cosine provide positive and negative factors that scale the sides of the right triangle.

9 of 11

Data Collection and Analysis�Area Measurement – two plots

Circle with Distance = 1 inch

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Inspired by Area the Easy Way

(X old, Y old)

North

Distance

(X new, Y new)

Heading

ΔX

ΔY

X new = X old + ΔX

​

Y new = Y old + ΔY

​

ΔX = Dist *Sine (Hdg)

ΔY = Dist *Cosine (Hdg)

Rectangle with Distance = 1 inch

10 of 11

Data Collection and Analysis�Area Measurement – two plots

Circle with Distance = 1 inch

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Inspired by Area the Easy Way

Rectangle with Distance = 1 inch

Red arrows indicate error accumulated through each run.

11 of 11

Data Collection and Analysis�Area Measurement