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Space Curves AML710 CAD LECTURE 11 SPACE CURVES Space Curves Intrinsic properties Synthetic curves SPACE CURVES A curve which may pass through any region of three- dimensional space, as contrasted to a plane curve which must lie on a single plane. Space curves are very general form of curves ¾ The generation of curves is a problem of curve fitting for given set of points or approximating a curve for these data points ¾ Curve Applications Many real engineering designs need curved mechanical parts, civil engineering designs, architectural designs, aeronautics, ship building ¾ Synthetic Curves The limitations of the analytic curves prompt us to study the synthetic curves 1 SPACE CURVES - Definitions ¾ Curve definition A continuously differentiable curve (function) is called smooth. To define the curvature it is convenient to use the Frenet frame, which is actually a pair of orthogonal coordinates having origin at the point of interest, P. n b P t Frenet Frame ¾ Curvature dt Frenet formulae = k(s).v(s).n(s) ds dn Binormal b=txn = −k(s).v(s).t(s) ds SPACE CURVES An example of a space curve A helix can be generated by the following parametric relations x = r cost y = r sin t z = bt r,b ≠ 0, − ∞ ≤ t ≤ ∞ 50 40 30 20 10 0 5 5 0 0 -5 -5 2 SPACE CURVES Exercise 1 A cubical parabola can be generated by the following parametric relations. Generate the space curve and also its components (x,t), (y,t) and (z,t). x = t y = t 2 z = t 3 0 ≤ t ≤1 SPACE CURVES Another example of a space curve The seam on a tennis ball can be generated by the following parametric relations π π x = a cos(θ + 4 ) − bcos3(θ + 4 ) π π y = asin(θ + 4 ) − bsin 3(θ + 4 ) z = csin(2θ ) c2 = 4ab, θ = 2πt;0 ≤ t ≤1 3 SPACE CURVES A conical helix can be generated by the following parametric relations with frequency a and height of the cone h h − z x = r cos(az) h h − z y = r sin(az) h z = z 0 ≤ z ≤ h FUNDAMENTAL THEOREM OF SPACE CURVES If two single-valued continuous functions k(s) (curvature) and t(s) (torsion) are given for s>0, then there exists exactly one space curve, determined except for orientation and position in space, where s is the arc length, k is the curvature, and t is the torsion. In other words, a relation f(k,t,s)=0 uniquely defines the space curve. The spherical curve taken by a ship which travels from the south pole to the north pole of a sphere while keeping a fixed (but not right) angle with respect to the meridians. The curve has an infinite number of loops since the separation of consecutive revolutions gets smaller and smaller near the poles. It is given by the parametric equations x = cost cosc y = sin t sin c z = −sin c c = tan −1(at) 4 Serret-Frenet Formulae for SPACE CURVES ⎡t ⎤ ⎡ 0 κ 0⎤⎡t ⎤ ⎢ ⎥ ⎢ ⎥⎢ ⎥ D⎢n⎥ = ⎢−κ 0 τ ⎥⎢n⎥ ⎣⎢b⎦⎥ ⎣⎢ 0 −τ 0⎦⎥⎣⎢b⎦⎥ Here all quantities are functions of s, the arc length which is a natural parameter for this situationt(s), n(s), b(s) are all functions of s. b = t × n Db = −τn Synthetic Curves When we combine polynomial segments to represent a desired curve, it is called a synthetic curve Generation ¾ Piecewise splines of low degree polynomials are combined to construct a curve ¾ Low degree polynomials both reduce the computational effort and numerical instabilities that arise with higher degree curves. ¾ However as low degree polynomials cannot span a large number of points, small segments of these curves are blended together to construct any desired curve in the practical design applications ¾ A common technique is to use series of cubic spline segments with each segment spanning only two points. ¾ Cubic spline is the lowest degree curve which allows a point of inflection and which has the ability to twist through space. 5 Types of Synthetic Curves 1. Cubic spline 2. Bezier Curve 3. B-spline Curve Local control Vs Global Control This aspect is application driven. How and where the slope, curvature etc are specified. Smoothness and Order of Continuity This gives an idea about the change of curvature ORDER OF CONTINUITY Curves are represented by joining segments of splines (piecewise polynomials) connecting them end to end. Therefore, the type or order of continuity becomes important for accepting them in design applications. The minimum continuity requirement is position continuity. This ensures the physical connectivity between different segments of the curve POSITION CONTINUITY – C0 CONTINUITY P3(t) Q1(t) P3(t) = Q1(t) Q (t) P2(t) 2 Q (t) P1(t) 3 6 SLOPE CONTINUITY – C1 CONTINUITY Q1(t) P3(t) P2(t) Q2(t) P (t) 1 Q3(t) P3 (t) = Q1(t) P3 '(t) = Q1'(t) CURVATURE CONTINUITY – C2 CONTINUITY Q (t) P3(t) 1 P2(t) Q2(t) P1(t) Q3(t) P3 (t) = Q1(t) P3 '(t) = Q1'(t) ″ ″ P3 (t) = Q1 (t) 7.
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