More on Construction of Surfaces

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1 More on Construction of Surfces Weihu Hong Mingshen Wu Nthn Borchelt 3 Deprtment of Mthemtics Clyton Stte University Morrow GA 3060 Deprtment of Mthemtics Sttistics nd Computer Science University of Wisconsin-Stou Menomonie WI Deprtment of Mthemtics nd Computer Science Western Crolin University Cullowhee NC 873 Astrct - Surfces cn e constructed y mny different methods such s revolving plnr curve out line in the sme plne or y shifting plnr curve long given vector (see [3). This pper presents other mens of construction of surfces: y moving stick y remodeling n existing surfce nd y inverting n existing surfce. Keywords: Surfce moving stick remodeling nd inverting. Introduction There re mny different surfces tht were given in mthemticl litertures (see [ [ [4). Two types of surfces: surfce of revolution nd surfce of cylinder were discussed in [3. In this pper we re interested in other mens of construction of surfces: y moving stick y remodeling n existing surfce nd y inverting n existing surfce. All our constructions cn e esily implemented vi computer lger system such s MthCAD [6 Mthemtic [7 Mple [8 nd MtL [9. The imges in this pper were creted y using MthCAD. Construction y Moving Stick Let us egin with γ ( which is smooth curve in spce. If α nd β re two linerly independent vectors such tht the curve does not lie in the spce spnned y α nd β then we re going to construct stick in the plne spnned y the vector α nd β s follows. Let v e unit vector in the spce spnned y α nd β. There exist two rel vlued functions c ( t ) nd c ( such tht v ( c( α + c ( β Define stick s follows: stick ( s sv( l. Imgine tht we re crrying the stick s we wlk long the curve. The stick is moving long the curve. The result of doing this will trce out strip which cn e expressed s follows. Theorem. Let c nd c e ny rel vlued functions such tht c ( + c ( v c ( α + c ( nd ( β stick ( s s v( l. Then the following will define strip long the curve γ ( with width of l. strip( γ ( + stick ( s where l Proof: For ech fixed vlue of the strip( represents line segment with length of l which is prllel to the vector v. γ ( is continuous on the intervl [ therefore strip ( is continuous on the [ [ l. Thus it defines strip. The proof is complete. Since the curve γ ( is smooth oth its tngent nd norml vectors re continuous nonzero vectors. Let τ ( α ( nd β ( e the unit tngen unit norml nd inorml vectors of γ ( respectively tht is τ ( β ( γ ( α( γ ( τ ( α(. τ ( α( τ ( τ ( Corollry. If the vectors α nd β re replced y the unit norml vector α( nd the unit inorml vector β ( t the point γ ( for ech in Theorem then the following defines strip. strip( γ ( + stick ( s where l Proof: Since oth the unit norml nd inorml vectors spn plne tht is perpendiculr to the tngent vector t ech vlue of therefore the stick will trce out strip long the curve. Exmple. Consider the fmous knot curve (see [) : γ ( t ) {(8 + 3cos(5)cos(t )(8 + 3cos(5)sin(t )5sin(5t )} where t [0π c / nd l then we will get the strip s shown in Figure. cos( c sin( nd l then we will get the strip s shown in Figure.

) nd l0.3 then we will get the fmous Möius strip (see [5) s shown in Figure 3 nd Figure 4. Fig. 7 (m5 n7) Fig.

5 We re to construct rion s follows: 3 Strt with the node curve y x + x tht looks like the Greek

3 Consider the curve in spce 3 γ ( t ) {cos( sin( cos(} where t [0π cos( t / ) c sin( t / ) nd

9 This hs the prmetriztion cos( cos( if π t 0 h( cos( cos( otherwise Define its 3D version s

5 define the rion s follows r r ion( γ ( + s( c j + ck ) then this will generte rion s shown in

4 Consider the sphericl spirl curve (see [): γ ( cos( mcos( n sin( m)cos( n sin( n 0π cos( t / )

2 Fig. Fig. Exmple. Consider the unit circle on the XY-plne: γ ( t ) {cos( sin( 0} where t [0π cos( t / ) c sin( t / ) nd l0.3 then we will get the fmous Möius strip (see [5) s shown in Figure 3 nd Figure 4. Fig. 7 (m5 n7) Fig. 8 (m 7 n ) Exmple.5 We re to construct rion s follows: 3 Strt with the node curve y x + x tht looks like the Greek chrcter α (see Fig. 9). Fig. 3 Fig. 4 Exmple.3 Consider the curve in spce 3 γ ( t ) {cos( sin( cos(} where t [0π cos( t / ) c sin( t / ) nd l0.5 then we will get nother Möius strip s shown in Figure 5 nd Figure 6. Fig. 9 This hs the prmetriztion cos( cos( if π t 0 h( cos( cos( otherwise Define its 3D version s follows γ ( { cos( h( 0} Let c cos( π cost / ) c sin( π cost / ) nd l0.5 define the rion s follows r r ion( γ ( + s( c j + ck ) then this will generte rion s shown in Figure 0 nd r r Figure. Where s usul j { 00} nd k {00 }. Fig. 5 Fig. 6 Exmple.4 Consider the sphericl spirl curve (see [): γ ( cos( mcos( n sin( m)cos( n sin( n 0π cos( t / ) c sin( t / ) nd l0. then we will get the strip s shown in Figure 7 nd Figure 8. Fig. 0 Fig. 3 Construction y emodeling In this section we discuss nother method of creting surfces: y remodeling n existing surfce. Firs we define remodeling trnsformtion of n existing surfce or curve. Definition. For given surfce nd model if there exists mpping such tht it will mp the surfce into the region tht is ounded y the given model then it is clled remodeling trnsformtion. The resulting surfce is clled remodeling trnsformtion surfce of the existing surfce.

Theorem. If model is given y nonnegtive function t [ c d nd f ( c d with f ( is given surfce then the surfce g ( f ( is remodeling trnsformtion surfce of f.

Thus the surfce f hs een mpped into region ounded y the umpy sphere of rdius m (. Therefore the surfce g ( f ( is remodeling trnsformtion surfce of f. The proof is complete. Exmple 3.

We define function s follows: m ( cos( ksin( k for integer k Then we pply the function to the unit sphere y selecting different vlues of k interestingly we will hve new surfces s

7 If we pply the function to the torus we will get the following new surfce s shown in figure 4. Fig.

But we define different function s follows π π Set δ. 8 0 u v) if u [(k ) k v [ jδ ( j + ) δ for k L8; j 0 L4; u v) 0.75 otherwise.

3 Theorem. If model is given y nonnegtive function t [ c d nd f ( c d with f ( is given surfce then the surfce g ( f ( is remodeling trnsformtion surfce of f. Proof: It is cler tht for ech point f ( on the surfce f the point g( will e ounded y umpy sphere of rdius m (. Thus the surfce f hs een mpped into region ounded y the umpy sphere of rdius m (. Therefore the surfce g ( f ( is remodeling trnsformtion surfce of f. The proof is complete. Exmple 3. We consider the unit sphere f ( {cos( cos( sin( cos( sin( } 0π 0 π. We define function s follows: m ( cos( ksin( k for integer k Then we pply the function to the unit sphere y selecting different vlues of k interestingly we will hve new surfces s shown in Figure. Exmple 3.3 We consider the torus: τ ( {cos( r cos( cos( sin( r sin( cos( r sin( } where r 0. which is shown s in figure 3. Fig. 3 We define function s follows kπ (k + ) π if t 6 6 (k + ) π (k + ) π r if t 6 6 k 0 L6 r 0.7 If we pply the function to the torus we will get the following new surfce s shown in figure 4. Fig. 4 Now if the function is pplied inside the torus s follows we will hve the new surfce s shown in figure 5. Fig. Exmple 3. We gin consider the unit sphere. But we define different function s follows π π Set δ. 8 0 u v) if u [(k ) k v [ jδ ( j + ) δ for k L8; j 0 L4; u v) 0.75 otherwise. We then pply the function to the unit sphere we get new surfce s shown in figure. u [ 0π v [0 π u [ 0π v [0.4 u [ 05.8 v [0 π Fig. ξ ( {cos( cos( cos( wherer 0. sin( sin( cos( sin( } Fig. 5 A remodeling trnsformtion cn lso e pplied to curve. The following exmple shows how this cn e used to crete some specil ojects. Exmple 3.4 We consider the unit circle γ ( t ) {cos( sin( 0} 0π We pply the modeling function defined in Exmple 3.3 to the circle denote the result f ( γ ( which is shown s in figure 6.

such tht dist ( O P) dist( O Q) r. The result of n inversion of given closed surfce is inside out nd outside in. Fig.

8 To mke the outside fce we define Outside( : f ( + s 00 0π 0 Likewise we define the inside fce Inside( : 0.5 f ( + s 00 0π 0 Which re shown s in figure 9 Fig.

Inversion in geometry is trnsformtion. Let P e given point. Let S e sphere centered t O nd rdius r.

To do this simply let x r cosθ y r sinθ z r cosϕ By sustitution F ( r cosθ y r sinθ sin ϕ z r cos 0 If the ove eqution is solved for r then r θ is otined. Theorem 3.

4 such tht dist ( O P) dist( O Q) r. The result of n inversion of given closed surfce is inside out nd outside in. Fig. 6 We re to use this to crete ger s shown in figure 7. Fig. 7 To do th we define the top nd ottom fces s shown in figure 8 s follows otto : s f ( 0π 0.5 nd top( : otto π 0.5 Fig. 8 To mke the outside fce we define Outside( : f ( + s 00 0π 0 Likewise we define the inside fce Inside( : 0.5 f ( + s 00 0π 0 Which re shown s in figure 9 Fig. 9 By ssemling these fces together the ger s shown in figure 7 is otined. 4 Construction y Inverting Lstly we discuss creting surfces y inverting n existing surfce. Inversion in geometry is trnsformtion. Let P e given point. Let S e sphere centered t O nd rdius r. The inverse of P with respect to S is point Q on the line OP For given surfce F(x y z) 0 we cn find its inversion surfce with respect to sphere centered t the origin with rdius of. To do this simply let x r cosθ y r sinθ z r cosϕ By sustitution F ( r cosθ y r sinθ sin ϕ z r cos 0 If the ove eqution is solved for r then r θ is otined. Theorem 3. The inversion surfce of the surfce F(x y z) 0 with respect to the sphere centered t the origin with rdius of cn e given s follows: InvS( { x( y( } where x( y( cosθ sinθ cosϕ Proof : Let Q e the point on the inversion surfce nd P e the corresponding point on the originl surfce with dist ( O P) θ It follows from the definition of inversion in geometry tht dist ( O Q) Thus the point Q hs coordintes { x( y( } where x( y( sinθ cosϕ The proof is complete. cosθ Exmple 4. Given the surfce (see [) x y + y z + z x find its inversion surfce with respect to the sphere centered t the origin with rdius of 3. It follows from theorem 3 tht 4 sin( [cos( + [sin(θ ) [sin( 4 nd the inversion surfce cn e expressed s follow 3cosθ sin ϕ 3sinθ sin ϕ 3cosϕ InvS ( θ θ

We hve oth the surfce nd its inversion surfce s shown in figure 0 nd figure respectively. 4 4 Fig. 0 Fig. Exmple 4.

It follows from theorem 3 tht θ cosθ + sinθ + cosϕ nd the inversion surfce cn e expressed s follow cosθ sin ϕ sinθ sin ϕ cosϕ InvS ( θ θ We hve oth the surfce nd its inversion surfce s shown in

This pper hs presented three different wys to construct surfce: y moving stick long curve y remodeling n existing surfce nd y inverting surfce with respect to sphere centered t the origin.

It is hoped tht these methods will inspire inquiry into dditionl methods of construction of surfces. 6 eferences [ Alfred Gry 993 Modern Differentil Geometry of Curves nd Surfces CC Press Inc. [ E.

5 We hve oth the surfce nd its inversion surfce s shown in figure 0 nd figure respectively. 4 4 Fig. 0 Fig. Exmple 4. Given the surfce x + y + z find its inversion surfce with respect to the unit sphere centered t the origin. It follows from theorem 3 tht θ cosθ + sinθ + cosϕ nd the inversion surfce cn e expressed s follow cosθ sin ϕ sinθ sin ϕ cosϕ InvS ( θ θ We hve oth the surfce nd its inversion surfce s shown in figure (the left grph is for the originl surfce s while the right grph is for its inversion). 5 Conclusions Fig. This pper hs presented three different wys to construct surfce: y moving stick long curve y remodeling n existing surfce nd y inverting surfce with respect to sphere centered t the origin. For ech method we hve given either useful formul or cler demonstrtion with mny exmples. These results might e used in surfce design for gmes or other industril purposes. It is hoped tht these methods will inspire inquiry into dditionl methods of construction of surfces. 6 eferences [ Alfred Gry 993 Modern Differentil Geometry of Curves nd Surfces CC Press Inc. [ E. Fer nd H. Huser Tody s Menu: Geometry nd esolution of Singulr Algeric Surfces Bulletin of the Americn Mthemticl Society Vol. 47 No. 3 July [3 W. Hong nd M. Wu 00 Construction of Surfces CGV00: -6 [4 Dvid H. von Seggern 007 CC Stndrd Curves nd Surfces with Mthemtic nd Edition Chpmn & Hll/CC [5 J. A. Thorpe 979 Elementry Topics in Differentil Geometry Springer-Verlg. 4 [6 [7 /index.html [8 [

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola.

k ) and directrix x = h p is A focal chord is a line segment which passes through the focus of a parabola and has endpoints on the parabola. Stndrd Eqution of Prol with vertex ( h, k ) nd directrix y = k p is ( x h) p ( y k ) = 4. Verticl xis of symmetry Stndrd Eqution of Prol with vertex ( h, k ) nd directrix x = h p is ( y k ) p( x h) = 4.