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Data Collection and Analysis Area Measurement How to measure the perimeter and area of an arbitrary closed shape?

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Data Collection and Analysis Area Measurement How to measure the perimeter and area of an arbitrary closed shape? Consider Descartes’ method: Could we do this by driving a robot around the shape? From Historical Connections in Mathematics, Volume III, Area the Easy WayArea 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. This is the polygon’s area.

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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 ΔXΔX ΔYΔY X new = X old + ΔX Y new = Y old + ΔY ΔX = Dist *Sine (Hdg) ΔY = Dist *Cosine (Hdg)

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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 ΔXΔX ΔYΔY 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.

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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.

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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. ΔXΔX ΔYΔY

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(X 1, Y 1 ) (X 0, Y 0 ) North ΔXΔX ΔYΔY (X 2, Y 2 ) Data Collection and Analysis Area Measurement Example path showing generated (x, y) pairs as the robot follows the line around. 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.

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(X 1, Y 1 ) (X 0, Y 0 ) North ΔXΔX ΔYΔY (X 2, Y 2 ) (X 3, Y 3 )(X 4, Y 4 ) (X 5, Y 5 ) Data Collection and Analysis Area Measurement Example path showing generated (x, y) pairs as the robot follows the line around. 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.

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Data Collection and Analysis Area Measurement – two plots Circle with Distance = 1 inch Inspired by Area the Easy Way (X old, Y old ) North Distance (X new, Y new ) Heading ΔXΔX ΔYΔY X new = X old + ΔX Y new = Y old + ΔY ΔX = Dist *Sine (Hdg) ΔY = Dist *Cosine (Hdg) Rectangle with Distance = 1 inch

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Data Collection and Analysis Area Measurement – two plots Circle with Distance = 1 inch Inspired by Area the Easy Way Rectangle with Distance = 1 inch Red arrows indicate error accumulated through each run.

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Data Collection and Analysis Area Measurement

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