Curves in space

A descriptive presentation on Curves and its features. An integral topic for mathematical methods in engineering.

1. Curves in Space Presented by- Piyali Dey (MEM20004) M.Tech in Mechanical Engineering, Tezpur University

2. OBJECTIVE Basic Notions of Vector Analysis Introduction to vector valued functions Curves and their tangents The length of a curve Curvature of a curve Torsion Twisting of curve in space and the Frenet frame Conclusion

3. Vector Analysis Vector can be defined a directed line segment having length and direction. Notations, basic terms and formulae of vector analysis to be used are: Cartesian Vector Representation V V z k y V j V x i x y z k i j 3 components in positive i, j, k directions Separating v magnitude and direction of each component vector will simplify the operation further in 3 dimensions considered Therefore, unit vectors

4. Projection of u V U x y z n= u nit vector perpendicular to the plane of u and v DOT PRODUCT produces a scalar quantity. The amount of V in U direction. CROSS PRODUCT produces a VECTOR quantity, orthogonal to both vectors. PROPER TI ES i.j=j.k=k.i=0 i.i=j.j=k.k=1 u.v=vu(commutative) V U x y =included angle z

5. S C A L A R T R I P L E P R O D U C T means the dot produc t of one of the vectors with the cross product of the other two vectors. The formula signifies the volume of the parallelepiped whose three edges denote three vectors, say, u, v, w Area Magnitude can be obtained by || V x U || D i s t r i b u t i v i t y S C A L A R & D O T P R O D U C T o f F O U R V E C T O R S with 4 vectors a, b, c, d D e r i v a t i v e s o f v e c t o r v a l u e d f u n c t i o n s :

6. Curves in space 2D Co -or dinate system W h a t a r e C U R V E S ? C U R V E R E P R E S E N T A T I O N Curve is a path of a point in motion It is 1D entity which can be represented by a single entity A curve can be represented parametrically by expressing the components of a vector from the origin to a point with coordinates x, y and z on it, as functions of a parameter r(t)=x(t) i + y(t) j + z(t) k W hy not? y=y(x), z=z(x) f(x,y,z)=0 g(x,y,z)=0 Explicit form Implicit form A straight line has a direction which we can describe by a unit vector in that direction: Thus the equations x = 2t, y = 3t, z = t describe a line that has the direction of the vector (2, 3, 1) and of the unit vector . ( 2,3,1)/73.74

7. Literature survey on Curve Representation Ron Goldman,2003, Pyramid Algorithm The paper discusses mainly about ambient space and coordinate systems. Representations of curve in computer graphics, geometric design: explicit, implicit, parametric, and procedural. Explicit representation: Considering, y = 3x + 1 represents a straight line, and y = x2 represents a parabola. Expressions of the form y = f(x) or z = f(x, y) are called explicit representations because they express one variable explicitly in terms of the other variables. But, Not all curves and surfaces can be captured readily by a single explicit expression. For example, the unit circle centered at the origin is represented implicitly by all solutions to the equation x2 + y2 − 1 = 0. If we try to solve explicitly for y in terms of x, we obtain which represents only the upper half circle. We must use two explicit formulas Implicit equations can be used to define closed curves and surfaces or curves and surfaces that self-intersect, shapes that are impossible to represent with explicit functions . Finding points on implicit surfaces f(x, y, z) = 0 can be even more formidable. Thus it can be difficult to render implicitly defined curves and surfaces. The parametric representation has several advantages. Like the explicit representation, the parametric representation is easy to render: simply evaluate the coordinate functions at various values of the parameters. Like implicit equations, parametric equations can also be used to represent closed curves and surfaces as well as curves and surfaces that self-intersect. In addition, the parametric representation has another advantage: it is easy to extend to higher dimensions.

8. Tangent of a curve T a n g e n t o f a c u r v e r(t+ t) r(t) r(t+ t)- r(t) r(t)=cos(t) i + sin(t) j + (t) k Here's an example: x = cos t, y = sin t, z = t. These particular equations describe the curve known as the "helix". Tangent vector- Where curve is going? We want to extract from our information about the curve, he intrinsic properties of the curve it represents. We further want to know how to compute them. T A N G E N T A general differentiable curve is one that at any point it has a slope and that slope will in general be in the direction of the vector.

9. Length of curve A r c L e n g t h o f a c u r v e The length of the curve over an interval [a,b] of the parameter t can be evaluated by breaking the interval in n segments and integrating it. R e p a r a m eritization We define one more parameter s(t) which represents the distance along the curve . The intrinsic information about the curve is contained in the relation between T(t) and s(t), between the tangent vector and the distance parameter along the curve. To a first approximation, the curve at any point is characterized by its slope there, which is the direction of T(t) or T(t(s)).

10. C u r v a t u r e o f a c u r v e The next important feature of interest is how much the curve differs from being a straight line at position s. We measure this by the curvature kappa (s), which is defined by which is, the magnitude of the change in unit tangent vector per unit change in distance along the curve. When parameterised by some t and not by arclength s, by chain rule: Curvy curve.

11. Normal Vector C o m p u t i n g C u r v a t u r e Since T is the unit vector in the direction of is the projection of a normal to v divided by the square of the magnitude of v. The center of curvature and the tangent vector to the curve, T(t), determine a plane called the plane of curvature. Since the radius of a circle is always normal to a vector tangent to it, a line from r(t) toward the center of curvature will be normal to T. The vector N(t), called the normal vector to the curve, is a unit vector pointing from r(t) toward the center of curvature Plane consisting is called osculating plane

12. Binormal Vector Torsion B(t), the "binormal vector" is a unit vector normal to both T and N, that is to the plane of curvature. By convention its direction is that of TxN. We define a(t) (the acceleration) to be the derivative with respect to t of v(t), (the velocity). In these terms, T is a unit vector in the direction of v, N is a unit vector in the direction of the projection of a normal to v, while B is a unit vector in the direction of v X a. B, N forms Normal plane B, T forms rectifying plane The "Serret Frenet Frame" defined by a curve of unit vectors, T(t), N(t) and B(t). The three vectors describin motion are Tangent Vector Normal Vector Binormal Vector We can go further. The next quantity of interest is how much the plane of curvature "twists". This is measured by the torsion of the curve, which is the magnitude of the derivative of the normal to the plane of curvature with respect to distance on the curve.

13. Twisting of curve Change in binormal vector will give us: Torsion

14. Conclusion Serret Frenet formulae: Intrinsic representation of a curve with The arc length parametrization of a curve i s completely determined by its curvature and torsion functions except for rigid bodies. We know

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