Introduction to virtual engineering Óbuda University John von Neumann Faculty of Informatics Institute of Intelligent Engineering Systems Lecture 3. Description of geometry László Horváth university professor

Parametric representation of curves Global and local characteristics of a curve Functions for the description of curves Methods for interpolation Bezier curves: characteristics, convex hull Curve: one-piece of piecewise (segmented)? Segments in a B-spline curve Characteristics of B-spline curve Control of a B-spline curve Contents László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

P (x,y,z) ( u ) u min u max Z X Y P General form of the parametric equation of curve: P(u)=[x(u) y(u) z(u)] where u min <= u <= u max The x, y, and z coordinates of model space point P in the function of the parameter u: x=x(u), y=y(u) és z=z(u) It defines points along a curve in the function of parameter u. It gives x,y, and z coordinate values at the point P for the parameter value u P of that point. P u is the position vector to point P. Parametric equation of a three dimensional curve Cartesian space Parametric representation of curves László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

Local characteristics at a point with parameter value u are: Tangent (t), Normal (n) Binormal (b) Curvature (r) t n b r Local characteristics t, n, and b define the accompanying trieder. Global characteristics: Control Degree class Global and local characteristics of a curve László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

Simple functions: analytical curves Polynomial: arbitrary analytical and free form shape Suitable for determination of tangents, normals, and curvatures. Differentiation of the function is easy. This class of functions is widely applied in description of geometry. General form of a polynomial of degree n is  i n i i xaxp    0 =  axaxaxaxp n n n n    … = Functions for the description of curves László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

Creating curve through points defined by experiments or calculations. Linear interpolation: straight segment between two points Arc through three points (quadratic analytical curve) Four points defne cubic curve. A Hermite interpolation: definition of a curve using two points and the demanded tangents at those points. Method by Hermite was applied by Ferguson and Coons. Interpolation polynomials were developed as solutions for the interpolation task. The most widely known method is the simple Lagrange polynomial for fitting curve through points. Methods for interpolation László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

First applications of mathematical descriptions of curves and surfaces were in design of cars and aircraft. Paul Bezier (Renault): introduced the control polygon, that controls the shape of curve by the position of control points. He applied Bernstein polynomials as blending (base) function. Concurently, de Casteljau (Citroen) developed similar method for the same purpose. Characteristics of the Bezier curve: Global control. Degree depends on the number or control points. Curve goes through the first and last control points. First and last segments of the control polygon are tangents of the curve. It is within the convex hull (see below). P 0 P 1 P 2 P 3 P 0 P 1 P 2 P 3 P 0 P 1 P 2 P 3 Bezier curves: characteristics, convex hull László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

One-piece curveBezier curve: global control. Change of position at a control point modifies the entire curve. Chain of Bezier curves. B-spline curve consists of segments. P i P i P i +1 i -1 segment i segment u 0 u 1 u 2 u 3 Curve: one-piece of piecewise (segmented)? László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

u= u 0 u 1 u 2 u 3 u 4 u 1 u 2 Knot Segment Parameter range of a segment Segments in a B-spline curve László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

Consists of segments. Continuity at segment borders. Local control. Spline base functions. Degree of the curve is same as degree of the base function. Different degree of segments is allowed. Curve goes through of the first and last control points only in case of special parameterization. Spline: Flexible steel ribbon in ship building. It was modeled. Characteristics of B-spline curve László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/

The B-spline base (blending) function is defined within a given parameter range. Example: Closed curve is controlled by six control points (vertices). It consists of six segments. Segment 1 is controlled by V0-V2. Segment 2 is controlled by V1-V3. And so on. V 0 V 1 V 2 V 3 V 4 V 5 Segment 1Segment 2 Control of a B-spline curve László Horváth ÓU-JNFI-IIES uni-obuda.hu/lhorvath/