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Problems
Consider an implicit surface
where
is a
polynomial in
. Consider the cube
and the part of
the surface inside this cube. The surface can be written in the
Bernstein basis as
Show the following properties.
The point
is on the surface if and only if
. What happens when
for all
?
Assuming the condition of question
is true, a necessary
and sufficient condition for the normal vector of surface
at
to be parallel to axis
is that
.
If
or if
for all
then there
is no piece of the surface in the cube under consideration.
Consider a cube
adjacent to the cube
. Within the new cube define another implicit polynomial
surface
of the same degrees in
as
.
Determine the conditions for the two surfaces to be position
continuous at the common face of the two cubes.
Following the condition of question
, determine the
conditions for the two surfaces to be tangent plane continuous at the
common face of the two cubes.
Show that the derivative of a Bézier curve (also called
hodograph) of the form:
is given by:
Sketch a cubic Bézier curve and its hodograph and their control
polygons.
The degree elevation formula for Bézier curves of degree
is given
(1.54).
Describe a process for approximating a Bézier curve of degree
with a Bézier curve of degree
using (1.54)
reversely.
Show how an explicit polynomial curve
, where
can be converted into a Bézier curve. Provide the control
points of the resulting Bézier curve. And
show how an explicit polynomial surface z = f(x,y), where
and
can be converted into a
Bézier patch and provide its control points.
Extend this to an explicit B-spline patch and provide its control
points.
Given a planar B-spline curve in the
plane with a non-uniform
knot vector, the
control polygon of which is symmetric with respect to the y-axis, find
if the curve is also symmetric about the y-axis.
What kind of curve is the result of a perspective projection
of an integral B-spline curve?
Consider the arc of the hyperbola
for
and revolve it around the axis
by
,
to obtain the quadrant of a surface of revolution, within the
first octant of the
coordinate system
. Express the resulting patch in terms of a rational biquadratic
parametric Bézier surface patch.
Examine what happens to a cubic B-spline curve in which two,
three, or four consecutive vertices, of its control polygon are
coincident.
Examine what might happen when a rational B-spline curve given by
(1.87) or a rational Bézier curve given by
(1.88) has weights some of which are positive and
some are negative. Examine the validity of the properties of the
rational B-spline or rational Bézier curves in such cases.
Make plots of the B-spline basis functions of the following
order
(degree
) and knot vector T:
n = 4,
n = 4,
n = 4,
n = 4,
n = 4,
n = 3,
n = 2,
Given a list of Cartesian points in 3-D space which represent a
non-periodic curve, construct a cubic
Bézier
curve using
least squares approximation of the points. Also, construct a cubic
B-spline curve with non-uniform knots using least squares
approximation of these points. A
user should be able to access your program with an arbitrary number of
points and coordinates of points. Include a simple visualization of
the results.
A cubic planar Bézier curve:
has the following control points
A designer decides to subdivide (split) the curve at
and
in order to be able to modify the curve in the interval
and
generate a particular shape feature required by his design.
Compute the coordinates of the control points of the three curve
segments generated by the above subdivision.
After the above subdivision, the middle segment of the three curve
segments created by subdivision is permitted to be modified by changing the
coordinates of its control points. Determine the conditions that the control
points must obey so that this curve segment maintains position continuity at
its ends with the end and the beginning of the other two curve segments.
Assuming position continuity is maintained as in (b), determine the
additional necessary conditions to maintain unit tangent vector continuity
at the ends of the middle curve segment.
Assuming conditions (b) and (c) are satisfied, determine the
coordinates of the vertices of the polygonal boundary of the convex hull
of the middle curve segment. You need to distinguish several special
cases.
Derive the monomial or power basis form of curve
of Problem 12 prior to any subdivision.
Derive the (uniform) B-spline form of curve
of Problem
12
prior to any subdivision. Make a plot of the curve
together with its Bézier and B-spline polygon illustrating the
principal features of the curve (such as tangencies at the ends etc.).
Compare the convex hulls of the curve in the Bézier and the
B-spline form in terms of the area they enclose (i.e. determine the
ratio of the two areas).
We are given a degree (2-1) integral Bézier surface patch
where the control points
are
Subdivide the surface patch into two patches by the
iso-parametric curve
and compute
the resulting control points of the two patches.
Consider the boundary curve of the patch
. Provide a tight upper bound for the maximum deviation of the
curve
in the interval
to the straight line passing via
and
for a fixed value of
, where
is a fixed positive integer.
Show that the given integral Bézier
surface is a developable surface.
Are there any umbilics on this patch?
Consider a curve
,
for
on a hyperbolic paraboloid
where
.
Compute the arc length of the curve on the hyperbolic paraboloid for
.
Compute the area of a region of the hyperbolic paraboloid bounded by
positive
axis,
and a parabola
.
Consider a torus parametrized as follows:
where
and
. Derive appropriate
formulae for the Gauss, mean and principal curvatures. Sketch the
torus and subdivide it into hyperbolic, parabolic and elliptic
regions. In a follow-up sketch illustrate the lines of curvature of
the torus. Explain the above subdivision and sketches.
Show that the curvature of a planar curve is independent of the
parametrization. Namely, if
is the curve, then a change of variables
does not affect the curvature.
Write a one-dimensional nonlinear polynomial solver based on Projected
Polyhedron algorithm. Use the solver to compute the roots of the degree 20
Wilkinson polynomial with different tolerances and discuss
robustness issues.
Convert an explicit curve
(
) into a
cubic Bézier curve.
Convert the following height function
into a bicubic Bézier patch.
Compute the characteristic points of the following curve
and trace it.
Consider the intersection
curve of a surface defined in Problem 15 with the plane
.
Derive an implicit equation
for this
intersection curve in the parameter space
. Find the
characteristic points of this curve, (border, turning, and singular
points).
Express this intersection curve as an explicit curve in the
parameter space. Indicate the resulting type and degree of
this curve. Sketch this curve in the parameter space
.
Prove that the above intersection curve is a planar rational
Bézier parametric curve of degree 4 in 3D space. Indicate how you
would compute its control points (but do not carry out the algebra).
Consider the following curves:
and
where
denotes the
th cubic Bernstein polynomial and
,
,
,
.
Compute all turning and singular points of
to the
highest possible accuracy, as well as the tangent lines at all these
points.
Using the results of (a) as a guide, sketch
.
Clearly indicate the turning and singular points on your sketch.
Compute the intersections of the two curves given above to
the highest possible accuracy. In addition to giving the Cartesian
coordinates of the intersection points, also include the parameter
values of the points and their multiplicity.
Write a program which determines all intersections of two
integral planar Bézier curves of arbitrary degrees
and
as
accurately as possible, given the control points of the two
curves. Your program should report the parametric values of the
intersection points as well as the Cartesian coordinates. Give four
examples to show how your program works.
The following three planar curves are given by:
Implicit
curve,
Cubic Bézier curve
where
and with
,
,
,
Cubic Bézier curve
where
and with
,
,
,
Compute the characteristic points of the first curve in the
rectangle [-5,5]
[-5,5] and trace it within the same rectangle.
Compute the intersections of the first and second curves,
and the second the third curves to the highest possible accuracy, and
identify their multiplicity.
Obtain a parametrization of the first curve in terms of
rational polynomials using the transform
. Illustrate
and
for all real
. Is this a good parametrization for computer
implementation (e.g. tracing of the curve) near
? Can you
suggest better parametrizations for the curve piece in the first
quadrant.
Compute the intersection curve between the bicubic Bézier patch
of Problem 21 and a plane
. Evaluate the
curvature of the intersection curve at
.
Give the implicit polynomial equation of a torus whose
section circle has radius 2, and whose center circle has radius 4
using the implicitization of a surface of revolution.
Assume the torus is situated so that it is centered at the origin and
the center circle lies entirely in the
-plane.
Using the implicit equation, compute all
intersections of the torus with the cubic Bézier curve having
control points
. Give both the Cartesian
coordinates of the intersection points and their associated parameter
values on the Bézier curve. Indicate which method you used to
solve this problem, and give all answers to at least 5
significant
digits.
Compute the minimum distance between a point (0.8, 0.7, 0.2) and
an iso-parametric line
of the bicubic Bézier patch of
Problem 21. Also compute the minimum distance between
the point and the bicubic patch.
Consider two planar Bézier curves which are cubic and
quadric with control points:
(0,0), (1, 1), (2,1), (3,0) and
(0,1), (
, -1), (
, 5), (
, -1), (3,0),
respectively. Compute all stationary points of their squared distance
function and classify them appropriately into extrema etc.
Identify the corresponding Euclidean distances, find the points of
intersection of the two curves and the angles between the tangents of
the two curves at the intersection points.
Consider an ellipsoid of revolution given by (3.81)
with
=
=1,
=2 and a cubic planar Bézier curve with control points (0,1, -2),
(0,0, -1), (0,0,1), (0,1,2) on the
=0 plane. Compute the
stationary points of the squared distance function between the
ellipsoid and the curve, classify them into extrema etc. Identify the
corresponding Euclidean distances, find the points of intersection and
the angles between the surface normals and the Bézier curve tangents
at the intersection points.
Find the stationary points of the squared distance function
between the plane z=0 and the wave-like Bézier surface patch of the
example in Sect. 8.5.4 and Fig. 8.11.
Classify the points into extrema etc., identify the corresponding
distances, and determine the intersections of the two surfaces.
Compare the locations of the above extrema with the locations of the
various curvature extrema in Sect. 8.5.4.
Consider a torus generated by revolving the circle
around the
axis by a full revolution.
Determine the stationary points of the squared distance function
between this torus and a) a plane
, b) a plane
, c) a plane
, d) a sphere with center the origin and radius
and e) a sphere with center the origin and radius
.
Consider a biquadratic Bezier surface patch
whose boundary eight
control points are coplanar so that the boundary curves form a square
on the
plane. The boundary non-corner control points
are in the middle of the corresponding boundary edges. The central
control point of the patch has coordinates (
,
,
) where
. Determine the surface unit normal
vector at the four corners, and at the center, and the extrema of the
Gauss, mean and principal curvatures and any umbilics as a function of
and illustrate this for
, 1, 10, 100. Sketch the
lines of curvature of the surface patch for these four values of
.
Find the range of mean curvature of a hyperbolic paraboloid
,
(bilinear surface), and
plot four levels of contour lines of mean curvature in the
-parameter space.
Given an implicit surface
= 0, formulate the problem of
tracing the lines of curvature and develop an algorithm to do this.
Test the resulting implementation for various standard algebraic
surfaces (quadrics, torii, cyclides).
Given an implicit algebraic surface
= 0, formulate the
problem of locating the umbilics of the surface (within a given
rectangular box with faces parallel to the coordinate planes).
Consider an ellipsoid
where
.
Show that umbilics
are located at
.
Show that the patterns of the four umbilics are of the lemon type.
Consider a degree (3-1) integral Bézier surface
where
=
,
=
,
=
,
=
,
=
,
=
,
=
,
=
.
Show that the Bézier surface is a developable surface.
Is there an inflection line? If so, find the
parameter
which contains the inflection.
Derive differential equations for geodesics
(10.17) - (10.20) on a parametric
surface using Euler's equation (10.24).
Write
a program which solves differential equations for geodesics
(10.17) - (10.20) as a boundary
value problem using a shooting method on a parametric surface.
For the surface patch of Problem 34
compute the geodesics between two diagonally opposite corners for
various values of
. How do these geodesics change as
changes from
0 to large positive values, e.g. in the interval [0,100]. What do you
expect in the limit
tends to plus infinity?
Let
be a planar, closed and convex curve ( e.g. a
circle, an ellipse, etc.) where the arc length
varies in the range
so that the length of the curve is
. Let
be its offset curve, where
is a positive distance and
is the unit normal vector of the curve
defined by
(see convention (b) of
(2.24) in Table 3.2.)
Show that the total length of the curve
exceeds the total length of the curve
by
.
Show that the area enclosed between the two curves is given
by
.
Show that the curvatures of the two curves are related by
where
is the curvature of
and
is
the curvature of the offset curve
.
Verify your results for questions a to c for a circle of radius
.
This problem focuses on the identification of cusps, extraordinary
points and self-intersections of offsets of planar curves (use
convention (b) of (2.24) for the
normal vector in Table
3.2.). Consider the ellipse
or
and its offset at ``distance''
, where
is
any real number.
Determine all the values of
for which there can be an
extraordinary point on some offset of the ellipse and the values of
at such points. Sketch the offsets at all such values of
.
For what range of values of
, are offsets of the ellipse
regular curves? Sketch a few such offset curves.
Determine a specific offset of the ellipse which includes several
cusps and self-intersections but no extraordinary points. Infinite such
cases exist. Give the parameter
values and coordinates of these cusps and self-intersections. (Hint:
Notice that self-intersections are on the axes of symmetry of the ellipse.)
Consider the planar cubic integral Bezier curve
with
control points
, and
. The
offset
of
at a distance
is given by
where
denotes the normal to
at the point
defined by
(see convention (b) of
(2.24) in Table 3.2).
Here
is called the
progenitor of
.
For values of
between 0
and some critical value
, the offset curve resembles its progenitor. At
,
however, the offset exhibits a cusp at a parameter value
because
. Compute the values of
and
.
Sketch the progenitor curve, and two offset curves,
one at a distance of
and one at a distance of around
.
The evolute of a planar curve is the curve of its center of
curvature. Show that cusps and extraordinary points of the offset lie
on the evolute. Illustrate the concept by examining the superbola
,
,
. Draw the curve, its evolute, and
several offsets with offset distance,
,
,
,
,
(all on the concave side).
Consider a pocket to be machined, bounded by the following
four curves:
1)
,
,
,
2)
,
3) Two circular blends of the first and second curves with radius,
.
Construct an approximation of the medial axis (skeleton) of the
pocket in terms of a set of linear spline curves. The skeleton is the
set of points inside the shape with two or more nearest points on the
boundary of the shape. The skeleton branches potentially start at the
curvature centers corresponding to points of maximum curvature of the
boundary. Next, you may specify the skeleton by writing differential
equations relating the tangent vector of the skeleton to known
functions. For simplicity, write these equations for the specific example.
Integrate these differential equations using the Runge-Kutta method.
Assuming you have cylindrical cutters with radii: 0.25, 0.5, 0.75,
, 2, describe an efficient method to accurately machine the pocket.
Write a program which approximates an offset curve of a planar rational
curve following the algorithm developed by Tiller and Hanson [421].
Give the implicit polynomial equation of a torus with axis in the
direction
(0,0,1), center circle radius
and section circle radius
where
. What
is the equation of the offset of this torus by
,
.
A pipe surface or canal surface of spheres of constant
radius is defined as the envelope of a family of spheres of constant
radius
whose centers describe a smooth curve, c(t) known as
the spine. Let
,
where
Show that the canal surface has an implicit equation
which results from eliminating
from the two
equations
and
.
Obtain the implicit equation of a torus by using the
approach of part a).
Show that canal surfaces can be obtained as generalized
cylinders by sweeping a circular cross-section along the spine.
In this problem we consider developing a blending surface
between the plane
and the right circular cylinder
. Because the cylinder is a surface of
revolution, we will, for simplicity, consider the cross-section of the
objects obtained by setting
. Our problem is to develop a
smooth surface between the cylinder and the plane by creating a
cross-section curve starting at height 1 on the cylinder and
terminating on the plane 2 units away from the origin.
As a first effort, consider using a quadratic Bezier curve
(i.e. a parabola) as a blend cross-section. The curve should have the
starting and ending points as indicated above, and to ensure a
smooth blend, the tangent to the curve at the start (end) point should
have the same direction as the tangent to the cylinder (plane). Give
the control points of this curve.
Using the results of part a), express the blending
surface (i.e. the surface of revolution characterized by the cross
section obtained in a) as rational B-spline (NURBS) surface.
Now suppose we want a cubic Bezier curve as the
cross-section of our blending surface. Give the control points of a
cubic Bezier curve which generates a ``good'' blend. To be ``good'',
the curve not only has to satisfy the boundary conditions indicated in
part (a), but also the area under the curve should be between
and
.
Now suppose we want to maintain curvature continuity at
the blending surface linkage curves in addition to position and
tangent plane continuity. Determine a sufficiently high degree Bezier
curve cross section to accomplish this.
Next: A. Color Plates
Up: Shape Interrogation for Computer
Previous: 11.6.3.4 A necessary and
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December 2009