AML710 CAD LECTURE 11 SPACE CURVES. Space Curves Intrinsic properties Synthetic curves

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1 AML7 CAD LECTURE Space Curves Inrinsic properies Synheic curves A curve which may pass hrough any region of hreedimensional space, as conrased o a plane curve which mus lie on a single plane. Space curves are very general form of curves The generaion of curves is a problem of curve fiing for given se of poins or approximaing a curve for hese daa poins Curve Applicaions Many real engineering designs need curved mechanical pars, civil engineering designs, archiecural designs, aeronauics, ship building Synheic Curves The limiaions of he analyic curves promp us o sudy he synheic curves

2 - Definiions Curve definiion A coninuously differeniable curve (funcion is called smooh. To define he curvaure i is convenien o use he Frene frame, which is acually a pair of orhogonal coordinaes having origin a he poin of ineres, P. n b P Frene Frame Curvaure Frene formulae Binormal bxn d k( s. v( s. n( s ds dn k( s. v( s. ( s ds An example of a space curve A helix can be generaed by he following parameric relaions x r cos y r sin z b r, b,

3 Exercise A cubical parabola can be generaed by he following parameric relaions. Generae he space curve and also is componens (x,, (y, and (z,. z y x Anoher example of a space curve The seam on a ennis ball can be generaed by he following parameric relaions ;, sin( sin ( sin( cos( cos( ab c c z b a y b a x

4 A conical helix can be generaed by he following parameric relaions wih frequency a and heigh of he cone h h z x r cos( az h h z y r sin( az h z z z h FUNDAMENTAL THEOREM OF If wo single-valued coninuous funcions k(s (curvaure and (s (orsion are given for s>, hen here exiss exacly one space curve, deermined excep for orienaion and posiion in space, where s is he arc lengh, k is he curvaure, and is he orsion. In oher words, a relaion f(k,,s uniquely defines he space curve. The spherical curve aken by a ship which ravels from he souh pole o he norh pole of a sphere while keeping a fixed (bu no righ angle wih respec o he meridians. The curve has an infinie number of loops since he separaion of consecuive revoluions ges smaller and smaller near he poles. I is given by he parameric equaions x cos cosc y sin sin c z sin c c an ( a

5 Serre-Frene Formulae for D n κ b κ τ τ n b Here all quaniies are funcions of s, he arc lengh which is a naural parameer for his siuaion(s, n(s, b(s are all funcions of s. b n Db τn Synheic Curves When we combine polynomial segmens o represen a desired curve, i is called a synheic curve Generaion Piecewise splines of low degree polynomials are combined o consruc a curve Low degree polynomials boh reduce he compuaional effor and numerical insabiliies ha arise wih higher degree curves. However as low degree polynomials canno span a large number of poins, small segmens of hese curves are blended ogeher o consruc any desired curve in he pracical design applicaions A common echnique is o use series of cubic spline segmens wih each segmen spanning only wo poins. Cubic spline is he lowes degree curve which allows a poin of inflecion and which has he abiliy o wis hrough space. 5

6 Types of Synheic Curves. Cubic spline. Bezier Curve. B-spline Curve Local conrol Vs Global Conrol This aspec is applicaion driven. How and where he slope, curvaure ec are specified. Smoohness and Order of Coninuiy This gives an idea abou he change of curvaure ORDER OF CONTINUITY Curves are represened by joining segmens of splines (piecewise polynomials connecing hem end o end. Therefore, he ype or order of coninuiy becomes imporan for acceping hem in design applicaions. The minimum coninuiy requiremen is posiion coninuiy. This ensures he physical conneciviy beween differen segmens of he curve POSITION CONTINUITY C CONTINUITY P Q P Q P Q P Q 6

7 SLOPE CONTINUITY C CONTINUITY P Q P Q P P ( Q ( P '( Q '( Q CURVATURE CONTINUITY C CONTINUITY P Q P P P ( Q ( P '( Q '( P ( Q ( Q Q 7