**Laboratory work. Preparation of NURBS templates
of analytical curves for CAD systems**

**Analytical curves**

Engineering
problems may require analytical curves with specific properties. Such curves can be derived analytically from the solution of specific
engineering problems.

A
designer’s powerful reserve is a whole palette of analytical curves, the
so-called, remarkable curves.

**NURBS curve patterns**

Currently,
CAD systems use NURBS curves as analytical curve templates (as a universal
internal representation of curves).

Second-order
curves are reduced to NURBS curves in an exact manner.

Spatial
spiral-helical lines and equidistants to flat curves are
approximated to NURBS curves.

When
approximating analytical curves with splines, you can use not only polylines of
base points, but also the differential characteristics of analytical curves at
these points. This approximation scheme is called
Hermite approximation.

For
geometrically stable approximation of analytical curves, the authors proposed a
method of approximation by a geometric spline in the NURBzS curve format [1].
The Hermite geometric determinant is used as initial
data - a reference polyline with fixed tangent and curvature vectors at the
points of the base polyline.

As an
example of approximation of an analytical curve, this work uses a clothoid
(Cornu spiral) [2]. The clothoid has a truly remarkable property: a linear law
of change in curvature along the length of the curve,
starting from zero. Clothoid splines of the 2nd order
of smoothness (splines composed of segments, circles and segments of a
clothoid) are widely used in the design of technical objects with functional
curves and surfaces.

It is also believed that the segment of an elastic bar
(physical spline) between two weights from a set of weights that fix the shape
of the physical spline has a linear law of change in curvature [2]. That is, it is accurately approximated by the segment of the clothoid.

The segment of the clothoid was used by the authors as a guide
curve for modeling the working surface of a plow moldboard according to the
diagram of Prof. Shchuchkin [3].

**Building a NURBS clothoid template. Methodical
instructions.**

Formula of
clothoid (Cornu spiral)

Let's
prepare a section of the clothoid on the segment 0<= t <= 3

Let's
calculate the first and second derivatives

Let's
calculate the curvature

Graph of
the clothoid site on the segment 0<= t <= 3

Clothoid
curvature graph

Let us
introduce the curvature function squared along the length of the curve

We
integrate the square of curvature along the length

So,
potential energy of the curve = 89.271

To
construct a NURBzS spline using the Hermite scheme, we prepare a reference
broken line of 21 points with fixed tangent vectors and curvature values

Base points

**Vectors of first derivatives (tangent vectors)**

Curvature

Values
are specified in the X coordinate

**Curvature vectors**

For a plane
curve, zero vectors are specified

Line graph
Curvature graph

Let's
prepare a table of clothoid parameters for the FairCurveModeler web application
on an Excel sheet

Insert >
Component > Excel > Microsoft Excel >

Create an
Empty Excel worksheet > Inputs - set to 12 > Outputs - set to 0 > In the placeholders, enter the variables

RX, RY, RZ,
RDX, RDY, RDZ, RKX, RKY, RKZ, RVKX, RVKY, RVKZ

Double
click on the table.

Select the
area with parameters on the Excel sheet that appears. Click PC mouse on sheet

In the
context menu, click Copy and then Copy

Transfer
the Excel table with parameters to AutoCAD / ZWCAD using the standard commands
of these systems and run the V_Hermite command.

**Bibliography**

1. Mufteev
V.G. High quality curve modeling based on the v-curve method. Applied geometry.
Applied Geometry [Electronic resource]: scientific. magazine
/ Moscow Aviation Institute (State Technical University) "MAI". -
Electron. magazine - Moscow: MAI, 2007. - No. 19; issue 9, -page 25-74. - Access mode to the journal:
http://www.mai.ru. - Cap. with title. screen. - State registration number 019164.

2. Fox A.,
Pratt M. Computational geometry. Application in design and production / Transl.
from English M.: Mir, 1982. -304 s.

3.
Mudarisov S.G., Mufteev V.G., Farkhutdinov I.M. Geometric modeling of dynamic
surfaces of working bodies of agricultural machines. Materials of the
All-Russian Scientific and Practical. Conf. "Current problems of the
agro-industrial complex", dedicated to. 65th anniversary of the Ulyanovsk State Agricultural Academy
and 20th anniversary of the Department of Life Safety and Energy February 6-8,
2008 - Ulyanovsk. pp. 136-143.