Stained Glass Design. Teaching Goals:

Transcription

1 Stined Glss Design Time required minutes Teching Gols: 1. Students pply grphic methods to design vrious shpes on the plne.. Students pply geometric trnsformtions of grphs of functions in order to design vrious shpes on the plne. 3. Students find equtions of ll curves obtined by geometric trnsformtions in generl form in either Crtesin or prmetric form s defined by the techer. 4. Students will determine domins of the independent vrible for the equtions of ll curves for the design of the shpe they construct in either Crtesin or prmetric form s defined by the techer. 5. Students will verify these results with the help of the softwre. Prior Knowledge Students should know grphs of liner functions nd the eqution of circle in Crtesin nd prmetric forms. Students should know the coordinte form of trnsltion by vector long ech coordinte xis. Students should know tht the eqution y = f(x ) corresponds to the trnsltion of the curve with the eqution y = f(x) by vector (, 0) long the x-xis. Students should know tht eqution y = f(x) + b corresponds to the trnsltion of the curve with the eqution y = f(x) by vector (0, b) long the y-xis. Students should know tht counter clockwise rottion hs positive ngle nd clockwise rottion hs negtive ngle, with ngles in the rnge [0, π]. Students should know tht the eqution y = - f(x) corresponds to the reflection of the curve with the eqution y = f(x) bout the x-xis. Students should know tht eqution y = f(-x) corresponds to the reflection of the curve with the eqution y = f(x) bout the y-xis. Problem: Construct stined glss design s shown in the picture below. This design is creted by using the curves given by the following equtions: y = x on [, 3] nd x + y = on [0, ]. Try to recrete similr design nd to crete other designs tht could be creted by these curves. 1

2 Determine equtions of ll curves on your design with their domin nd verify them with the help of the softwre. Comment: the ide of this problem is to be ble to send this informtion to nother person who should be ble to recrete this design bsed on the informtion provided (not by mindless copying). Prt 1 Setting Up Problem in Geometry Expressions 1. Open new file. If xes do not pper in the blnk document, crete them by clicking the Toggle grid nd xes icon on the top toolbr.. Select Function from the Drw toolbox. In the Type: box click the down rrow on the right nd select Prmetric from the list. In the next row, X=, enter: T ; in the next row, Y=, enter: bs(t); for Strt: enter: ; for End: enter: Click the eqution of the function, right-click nd select Hide from the context menu. Select the curve, right-click nd select Properties from the context menu. Click the Line Style row nd it s down rrow to chnge it to Dot. 4. Select Function from the Drw toolbox. In the Type: box click the down rrow on the right nd select Prmetric from the list. In the next row, X=, enter: *cos(t) ; in the next row, Y=, enter: *sin(t); for Strt: enter: 0; for End: enter: Click the eqution of the function, right-click nd select Hide from the context menu. Select

3 the curve, right-click nd select Properties from the context menu. Click the Line Style row nd it s down rrow to chnge it to Dot Prt Creting Stined Glss Design Q1: How cn you construct squre from the given curves? A: It is expected tht students will use trnsltion long y-xis nd reflection bout x-xis ccording to the following steps, for exmple: 1. Click the stright line function nd select Trnsltion from the Construct toolbox. Construct vector of trnsltion from the origin down long the y-xis. The trnslted function will pper on the screen.. Select the end point of the trnsltion vector nd click Coordinte in the Constrin toolbox. Enter (0,) for the coordintes of the endpoint of the trnsltion vector. 3. Select the trnsltion vector, right-click nd select Hide from the context menu. 4. Select the trnslted function. Right-click nd select Properties from the context menu. Chnge the Line Color to Blue nd the Line Style to Solid 3. 3

4 5. Click the trnslted function nd select Reflection from the Construct toolbox. Click the x- xis nd the function will be reflected over the x-xis. Adjust the line color nd style s in step 4 bove A B (0,) Q: How cn you construct qurter circle with its center t the bottom vertex of the squre? A: Students my suggest rotting the given qurter circle by 4 π bout the origin nd then trnslting the center of the circle down to the point (, 0). Here re the steps: 1. Select the rc nd click Rottion from the Construct toolbox. Click the origin nd the box will pper for entering the ngle. (If the Symbols toolbox is not displyed, click View / Tool Pnels / Symbols) π cn be found in the lower right corned of the dilog. Enter π/4.. Select the ngle of rottion lbel nd Hide it. 3. Select the rotted rc nd click Trnsltion from the Construct toolbox. Construct the trnsltion vector from the origin to the point (0,). Hide the intermedite copy of the rc nd the trnsltion vector. 4. Chnge the new rc s line color to Red nd the line style to Solid 3. Q3. How cn you construct qurter circle with the center in the top vertex of the squre? 4

5 A: Students my suggest similr steps s bove, but the simplest wy to do this is to reflect the newly constructed rc bout the x-xis. The following steps will ccomplish this: 1. Click the rc with its center t the bottom vertex of the squre. Select Rottion from the Construct toolbox nd select the x-xis. The imge will pper on the screen.. Select the rc nd right-click Properties to chnge the line color to Red nd the line style to Solid A B (0,) Q4: How cn you construct qurter circle with the center in the left vertex of the squre? A: Students my suggest rotting the given qurter circle by -π/4 bout the origin nd then trnslting the center of the circle down to the point (, 0). Here re the steps: 1. Select the originl rc nd click Rottion from the Construct toolbox. Click the origin nd the ngle edit box will pper. From the Symbols toolbox, find π nd type: -π/4.. Select the ngle of rottion lbel nd Hide it. 3. Select the rotted rc nd click Trnsltion from the Construct toolbox. Construct the trnsltion vector from the origin to the point (, 0). Hide the intermedite copy of the rc nd the trnsltion vector. 4. Select the rc nd right-click Properties to chnge the line color to Red nd the line style 5

6 to Solid 3. Q5. How cn you construct qurter circle with the center in the right vertex of the squre? A: Students my suggest similr steps s bove, but the simplest wy to do this is to reflect the rc with the center t the left vertex bout the y-xis. The following steps will ccomplish this: 1. Click the rc nd select Reflection from the Construct toolbox. Click the y-xis nd the imge will pper on the screen.. Select the imge nd right-click Properties to chnge the line color to Red nd the line style to Solid Click-nd-drg the originl qurter circle rc to djust the trnsformed rcs to the desired size D (,0) - A B (0,) Prt 3 Finding Equtions of the Curves in the Stined Glss Design In this prt of the problem students hve to determine the eqution of ech curve or function produced by trnsformtion of the originl curves. Bsed on the students prior knowledge, the techer cn determine in which form students will write the equtions of the curves: Crtesin or prmetric. Students should lso determine the domin of the independent vrible for the 6

7 equtions of ech curve. This should be completed independently nd then verified with the help of the softwre. A. A Squre is formed by two curves: y = x 3, [ 3,3] produced by the eqution y = x, [ 3,3] trnslted by vector (0, - 3) nd y = x + 3, [ 3,3] produced by eqution y = x, [ 3,3] reflected bout x-xis nd trnslted by vector (0, 3). The inner design is mde of four qurter circles. When the circles re tngent, the rdius of the circle is equl to hlf of 3 the side of the squre, =, nd the x-coordintes of the points where the rcs intersect the squres re.5 nd 1.5. The student cn crete the equtions of the rcs s bounded circles centered t the origin, trnslted to the vertices of the squre. Thus, the equtions of the rcs of the circles follow. At the bottom vertex (0,): the circle 9 x + y =, trnslted by vector (0,) becomes 9 x + ( y+ 3) =, on [.5, 1.5] ; t the top vertex (0, 3): the bottom rc reflected bout x-xis 9 9 becomes x + ( y 3) = on [.5, 1.5]; t the left vertex (, 0): the circle x + y =, 9 trnslted by vector (,0) becomes ( x+ 3) + y =, on [ 1.5, ] ; nd t the right 9 vertex (3, 0): the left rc reflected bout y-xis becomes ( x 3) + y =, on [3 1.5,1.5]. Students my hve difficulty finding the domins of the functions. Prt 4 Verifiction of the Equtions with Geometry Expressions Here re the steps to find the equtions using Geometry Expressions: 1. Select ech curve, one t time, nd choose Implicit Eqution from the Clculte toolbox. The eqution of the curve will pper on the screen. 7

8 9+X -6 Y+Y - =0 3 Y=- + X 9+6 X+X +Y - = X+X +Y - =0-5 D (,0) - A Y=+ X - B (0,) 9+X +6 Y+Y - =0 Students should substitute the vlue of nd verify tht the equtions given by the softwre re the sme s ones they derived. In order to verify points of intersections, students cn construct squre nd circle in new document nd determine points of intersection using the following steps: 1. Open new file. If xes do not pper in the blnk document, crete them by clicking the Toggle grid nd xes icon on the top toolbr.. Select Point from the Drw toolbox nd construct four points for the vertices of the squre. 3. Select one of the points, choose Coordinte from the Constrin toolbox nd type (0, 3). Constrin the other points with the coordintes (0, ), (3, 0) nd (, 0). 4. Select Line Segment from the Drw toolbox nd connect the points to construct squre. 5. Select Circle from the Drw toolbox nd construct circle with the center t the point (3, 0). With the circle selected choose Rdius from the Constrin toolbox nd enter for the vlue. 6. Holding the CTRL key, select the circle nd segment in the 1 st qudrnt. Choose Intersection from the Construct toolbox. The point of intersection will pper. 7. Select the intersection point, choose Coordinte from the Clculte (Symbolic) toolbox nd the coordintes of the intersection point will be displyed. 8

9 B 4 3-, (0,3) G -6 C (,0) - F A (3,0) H (0,) D Comment: students cn substitute the vlue of nd find the other intersection points bsed on symmetry. Extensions 1. Students cn consider the following cses for the stined glss designs 3. when < (rcs do not intersect) b. when 3 < < 3(ech pir of rcs intersect) c. when > 3. Investigte cse b where ech pir of rcs intersects. Try to mke the sum of the res of the qudrilterl formed outside the rcs nd the regions formed by the overlpping sectors in given proportion, for exmple, 1:4 to the totl re of the squre. This problem should be solved by pproximtion, using squre to pproximte the shpe formed in the center outside the rcs. 9

10 Students cn use circles nd segments nd construct prt of the design in new file nd then find coordintes of the intersection point for the vertex of the centrl squre in terms of prmeter , AJ' - N' AF O' AI' P 4 6 V' AF' S' - AH' 3. 10

11 3. Investigte the cse c where > 3. In this cse concve qudrilterl shpes re formed t the vertices of the squre with convex qudrilterl t the origin. Students cn gin use circles nd segments, construct the design nd pproximte the curved shpes with squres to look for specific rtio of res for the design ,

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