LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= NAME:

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Type your name (and your lab partner if working with someone) at the top. Save this to your H drive as LASTNAME_LAB5.mw Save often while you are working. Make sure you answer all questions!!! When you are done, email me your lab beaverc@wou.edu The lab is due Friday 11/9

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= If x and y are both given as functions of a third variable, say t, then we have a set of equations parametrized by t and we call them parametric equations: x = f(t), y = g(t). Each value of t determines a point (x,y), which we can plot. As t varies, the points (x,y) trace out a curve called a parametric curve. It is often convenient to think about t as time and as time varies your point travels along your curve. As an example, suppose we have 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, we can graph it below to see what it looks like as t varies from 0 to 10: 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

Lines through two given points are particularly easy to parametrize. Suppose we want the line through points P1=(x1,y1) and P2=(x2,y2). If we let 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 then at t=0, we have (x,y)=(x1,y1)=P1 and at t=1 we have (x,y)=(x2,y2)=P2 so we get the line joining P1 and P2. If we restrict t to 0 to 1, then we get the line segment joining P1 and P2. Example: The equation of the line joining P=(1,2) and Q=(3,5) is given by 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

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYiIiJGKiomIiIjRipGJEYqRipGJUYlRiU=

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

Zio2I0kidEc2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlLCYiIiMiIiIqJiIiJEYrRiRGK0YrRiVGJUYl

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Let's graph it to check: 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

QUESTION 1: What is the set of parametric equations defining the line between P=(-2,5) and Q=(3,4)? Use my command lines above to define x(t) and y(t) ( by changing the appropriate numbers) and plot the graph to see if you're right. ANSWER 1: The answer to the question is the graph above showing the line through P=(-2,5) and Q=(3,4).

The derivative of a parametric equation is found by using the Chain Rule. Let x = f(t) and y = g(t), then if f and g are differentiable functions the chain rule gives: 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 So we can solve for LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2I1EhRictRiM2JC1JJm1mcmFjR0YkNigtRiM2JC1GLDYlUSNkeUYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy9GPVEnbm9ybWFsRictRiM2JC1GLDYlUSNkeEYnRjlGPEY/LyUubGluZXRoaWNrbmVzc0dRIjFGJy8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0ZLLyUpYmV2ZWxsZWRHUSZmYWxzZUYnRj9GK0Y/ 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Let's check this by using it to find the derivative of our line through P=(1,2) and Q=(3,5) given by x = 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 QUESTION 2: Find dy/dx on paper using the chain rule.(dy/dx = y'(t)/x'(t)) Since this is the paramatrization of a line, the derivative should equal the slope. Compute the slope directly using P and Q and check that it matches the derivative. Type in your work below: ANSWER 2:

Bezier Curves are used in computer aided design and are named after the French mathematician Pierre Bezier (1910-1999), who worked in the automotive industry. A cubic Bezier curve is determined by four control points, P0=(x0,y0), P1=(x1,y1), P2=(x2,y2), and P3=(x3,y3) , and is defined by the parametric equations LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY9LUkjbWlHRiQ2JVEieEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIj1GJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC1GLDYlUSN4MEYnRi9GMi1GNjYtUScmc2RvdDtGJ0Y5RjtGPkZARkJGREZGRkgvRktRJjAuMGVtRicvRk5GVi1JJW1zdXBHRiQ2JS1JKG1mZW5jZWRHRiQ2JC1GIzYmLUkjbW5HRiQ2JFEiMUYnRjktRjY2LVEoJm1pbnVzO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EsMC4yMjIyMjIyZW1GJy9GTkZiby1GLDYlUSJ0RidGL0YyRjlGOS1GIzYkLUZbbzYkUSIzRidGOUY5LyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1GNjYtUSIrRidGOUY7Rj5GQEZCRkRGRkZIRmFvRmNvRmlvRlItRjY2LVEifkYnRjlGO0Y+RkBGQkZERkZGSEZVRlctRiw2JVEjeDFGJ0YvRjJGUkZkb0ZSLUZZNiVGZW4tRiM2JC1GW282JFEiMkYnRjlGOUZccEZfcEZpb0ZSLUYsNiVRI3gyRidGL0YyRlItRlk2JUZkb0ZqcEZccEZSRmVuRl9wLUYsNiVRI3gzRidGL0YyRlItRlk2JUZkb0Znb0ZccEY5 y =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 where 0<= t <= 1. Notice that when t=0 we have (x,y)=(x0,y0) and when t=1 we have (x,y)=(x3,y3), so the curve starts at P0 and ends at P3. The Bezier Curve can be defined by x=f(t), and y=g(t) where f(t) and g(t) are the functions below where P0=(x0,y0), P1=(x1,y1), P2=(x2,y2), P3=(x3,y3) are the four control points. We'll enter the functions leaving the coordinates as variables so we can change them later without retyping the whole function. (Hit enter on all lines I've entered for you as you go) 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZALUkjbWlHRiQ2JVEiZ0YnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RKiZjb2xvbmVxO0YnL0YzUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGPS8lKXN0cmV0Y2h5R0Y9LyUqc3ltbWV0cmljR0Y9LyUobGFyZ2VvcEdGPS8lLm1vdmFibGVsaW1pdHNHRj0vJSdhY2NlbnRHRj0vJSdsc3BhY2VHUSwwLjI3Nzc3NzhlbUYnLyUncnNwYWNlR0ZMLUYsNiVRInRGJ0YvRjItRjY2LVEoJnNyYXJyO0YnRjlGO0Y+RkBGQkZERkZGSC9GS1EmMC4wZW1GJy9GTkZWLUYsNiVRI3kwRidGL0YyLUY2Ni1RJyZzZG90O0YnRjlGO0Y+RkBGQkZERkZGSEZVRlctSSVtc3VwR0YkNiUtSShtZmVuY2VkR0YkNiQtRiM2Jy1JI21uR0YkNiRRIjFGJ0Y5LUY2Ni1RKCZtaW51cztGJ0Y5RjtGPkZARkJGREZGRkgvRktRLDAuMjIyMjIyMmVtRicvRk5GaG9GTy8lK2V4ZWN1dGFibGVHRj1GOUY5LUYjNiUtRmFvNiRRIjNGJ0Y5RmpvRjkvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY2Ni1RIitGJ0Y5RjtGPkZARkJGREZGRkhGZ29GaW9GXnBGZW4tRiw2JVEjeTFGJ0YvRjJGZW5GT0Zlbi1GaW42JUZbby1GIzYlLUZhbzYkUSIyRidGOUZqb0Y5RmFwRmRwRl5wRmVuLUYsNiVRI3kyRidGL0YyRmVuLUZpbjYlRk9GXHFGYXBGZW5GW29GZHAtRiw2JVEjeTNGJ0YvRjJGZW4tRmluNiVGT0ZccEZhcC1GNjYtUSI7RidGOUY7L0Y/RjFGQEZCRkRGRkZIRlVGTUZqb0Y5 The general formula for the parametric equation for the line segment joining P0 and P1 (we'll call it line01) is given by x=line01x(t) and y=line01y(t) as defined below: 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 Similarly, the parametric equation for the line segment joinging P1 and P2 (line12) is given by x=line12x(t) and y=line12y(t): 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 And the parametric equation for the line segment joinging P2 and P3 (line23) is given by x=line23x(t) and y=line23y(t): 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 We want to graph the curve where our control points are P0=(4,1); P1=(28,48); P2=(50,42); and P3=(40,5). We can assign our general coordinates to these values using the syntax x0:=4, y0:=1, etc (note the colon followed by the equal sign - very important!). Do this. JSFH JSFH JSFH JSFH JSFH JSFH JSFH JSFH Now that we've assigned these values, the variables in the equations we defined above should have been replaced by these values. For example pressing return below should yield: 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEiZkYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUYsNiVRInRGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA PLOTTING: To plot a parametric curve we need a function for x, a function for y, and a range for t. We enter these into plot with square brackets around them: For example Plot([x(t),y(t),t=0..1]) Let's plot our Bezier function. Check the the graph starts at the point P0=(4,1) and ends at the point P3=(40,5). 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

Now we'll plot the line segments joining the pairs of control points on the same axis as the Bezier function. To plot multiple parametric functions we begin with a curly bracket { type each function enclosed with square brackets, separate the functions with commas, ending with a matching curly bracket } Here's the syntax: plot({[x1(t),y1(t),t=0..1],[x2(t),y2(t),t=0..1]}); Here's the command to plot our line segements with our curve: LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVElcGxvdEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYlLUY2NiYtRiM2Ky1GNjYmLUYjNi8tRiw2JVEiZkYnRi9GMi1GNjYkLUYjNiUtRiw2JVEidEYnRi9GMi8lK2V4ZWN1dGFibGVHUSZmYWxzZUYnL0YzUSdub3JtYWxGJ0ZPLUkjbW9HRiQ2LVEiLEYnRk8vJSZmZW5jZUdGTi8lKnNlcGFyYXRvckdGMS8lKXN0cmV0Y2h5R0ZOLyUqc3ltbWV0cmljR0ZOLyUobGFyZ2VvcEdGTi8lLm1vdmFibGVsaW1pdHNHRk4vJSdhY2NlbnRHRk4vJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR1EsMC4zMzMzMzMzZW1GJy1GLDYlUSJnRidGL0YyRkVGUUZJLUZSNi1RIj1GJ0ZPRlUvRlhGTkZZRmVuRmduRmluRltvL0Zeb1EsMC4yNzc3Nzc4ZW1GJy9GYW9GW3AtSSNtbkdGJDYkUSIwRidGTy1GUjYtUSMuLkYnRk9GVUZpb0ZZRmVuRmduRmluRltvL0Zeb1EsMC4yMjIyMjIyZW1GJy9GYW9GX28tRl5wNiRRIjFGJ0ZPRkxGT0ZPLyUlb3BlbkdRIltGJy8lJmNsb3NlR1EiXUYnRlEtRjY2Ji1GIzYvLUYsNiVRKGxpbmUwMXhGJ0YvRjJGRUZRLUYsNiVRKGxpbmUwMXlGJ0YvRjJGRUZRRklGZm9GXXBGYXBGZ3BGTEZPRk9GanBGXXFGUS1GNjYmLUYjNi8tRiw2JVEobGluZTEyeEYnRi9GMkZFRlEtRiw2JVEobGluZTEyeUYnRi9GMkZFRlFGSUZmb0ZdcEZhcEZncEZMRk9GT0ZqcEZdcUZRLUY2NiYtRiM2Ly1GLDYlUShsaW5lMjN4RidGL0YyRkVGUS1GLDYlUShsaW5lMjN5RidGL0YyRkVGUUZJRmZvRl1wRmFwRmdwRkxGT0ZPRmpwRl1xRkxGT0ZPL0ZbcVEifGZyRicvRl5xUSJ8aHJGJ0ZMRk9GTy1GUjYtUSI7RidGT0ZVRldGWUZlbkZnbkZpbkZbb0Zdb0ZccEZMRk8= JSFH From the graph above, it appears that the tangent at P0 passes through P1 and the tangent at P3 passes through P2. To prove this we need to find the derivative of the Bezier curve so we can compute the slope of the tangent line. Recall from above that the chain rule tells us that 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 JSFH So to get our derivative we want top/bottom: 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 JSFH The point P0 occurs at t=0, so to find the slope, we plug in t=0. To do this we substitute 0 in for t using the subs command: 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 JSFH QUESTION 3: Use this slope to find the equation of the tangent line at P0 and determine if this line goes through. P1. Explain your answer below: ANSWER 3: QUESTION 4: The point P3 corresponds to t=1 on the curve. Repeat the process above, substituting t=1 into the derivative to find the slope at P3 and determine if this line goes through P2. JSFH JSFH JSFH JSFH ANSWER 4: If we vary the control points, we can produce Bezier curves with various shapes. The curve always starts at P0, heads off in the direction of P1, then P2 and ends at P3. Try to produce a Bezier curve with a loop by varying P1=(x1,y1). To do this, just reassign x1 and y1 new values (your equations should automatically update). Start by choosing any values to see what happens, then modify to get your loop. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjeDFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEjeTFGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSSNtb0dGJDYtUSomY29sb25lcTtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRj0vJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTC8lK2V4ZWN1dGFibGVHRj1GOQ== Now plot the functions again to see how things changed. (just hit enter on the plot command line below) QUESTION 5: Keep playing with x1 and y1 until you get a graph with a loop. ANSWER 5: Just leave the graph below so I can see you got it. 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

JSFH JSFH JSFH JSFH QUESTION 6: Some laser printers use Bezier curves to represent letters and other symbols. Experiment with the control points (by reassigning the x0,y0,x1,y1,etc. values until you find a Bezier curve that gives a reasonable representation of the letter C. ANSWER 6: Your graph will be your answer to this question. JSFH JSFH JSFH JSFH JSFH JSFH Here's the plot command so you can test out how varying your control points changes the curve. 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=