3. cad dr.c.k.shene geometric concepts

8/10/2019 3. CAD Dr.C.K.shene Geometric Concepts

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Unit 2: Geometric Concepts

Coordinate Systems, Points, Lines and Planes

Two-Dimensional Objects

Points

Thexy-coordinate plane has two coordinate axes, thex- andy-axis. They are perpendicular to each

other. Non-perpendicular axes can be used; but, the computation cost is higher.

A point in thexy-plane is represented by two numbers, (x, y), wherexandyare the

coordinates of thex- and y-axes.

Lines

A line in thexy-plane has an equation as follows:

Ax + By + C= 0

It consists of three coefficientsA,Band C. Cis referred to as the constant term. IfBis non-zero, the

line equation can be rewritten as follows:

y = m x + b

where m = -A/Band b = -C/B. This is the well-known slope-intercept form in which mand bare the

slopeand the intercept(i.e., the intersection point of the line and they-axis). IfBis zero, the line

equation becomesAx + C= 0, which is a line parallel to they-axis and intersects thex-axis at point

(0,-C/A).

The line equation has three coefficients; but, there are only two independentones. That is,

given a line equation, dividing the equation by one of its non-zero coefficients will not

change the line. For example, the line equation 4x+5y+7=0 is equivalent tox+1.25y+1.75=0

and 0.8x+y+1.4=0. Dividing an equation by a non-zero constant is usually referred to asnormalization. There is one normalization that is very important to us. It is used to compute

the distance between the origin to a line.

Suppose the line equation isAx+By+C=0, the distance between the origin and the line is

given as follows:


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Thus, after normalizing the line equation by dividing it with the square root of the sum of the

squares ofAandB, the absolute value of the new constant term is the distance between the

origin and the given line.

Parallel and Perpendicular Lines

Given two lines as follows:

Ax + By + C= 0

Ex + Fy + G= 0

they are parallel to each other if their slopes are equal. Since the slopes are -A/Band -E/F(ifBandF

are both non-zero), two lines are parallel to each other if and only if the following holds

AF=BE

Note that we can assume thatBorFare both non-zero. Otherwise, if both are zero, the lines areparallel to they-axis and hence are parallel to each other.

Two lines are perpendicular if and only if the product of their slopes is -1. With the line

equations above, two lines are perpendicular to each other if and only if the following holds:

AE=-BF

Three-Dimensional Objects

Points

The coordinate system in space needs three coordinate axes, thex-,y- andz-axis. Therefore, a point in

space has three components (x,y,z), wherex,yandzare the coordinates of thex-,y- andz-axis.

Planes

A plane in space has an equation as follows:

Ax + By + Cz + D= 0

It consists of four coefficientsA,B, CandD, whereDis the constant term. Similar to the line case,

the distance between the origin and the plane is given as

The normal vector of a plane is its gradient. The gradient of an equation f(x,y,z)=0 is defined

as follows:


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For a plane, its normal vector is simply .

Unfortunately, a line in space cannot be represented with a single equation. However, it can

be considered as the intersection of two planes. To ease our discussion, we shall switch to

using vectors, although this traditional notation is still very useful.

Vectors

Vectors have a very important advantage in geometric computing, because it is "coordinate free." The

meaning of "coordinate free" will be clear in later discussions. All vectors will be in boldface like aand A.

A vector is similar to a point. If it is a vector in the plane (resp., space), it has two (resp.,

three) components. Thus, a vector in a n-dimensional space has ncomponents. For our

applications, we shall distinguish two types of vectors:positionvectors and directionvectors.

A position vector gives the position of a point. More precisely, a point is a vector. A direction

vector gives a direction. Hence, it is not a point. In what follows, position vectors and

direction vector are written with boldface upper case and lower case, respectively. For

example, Aand aare position and direction vectors, respectively. In many cases, such

distinction is unnecessary.

As you have learned in linear algebra or in calculus, you can add and subtract vectors; but

you can only multiply or divide a vector with constants.

The length of a vector is the square root of the sum of squares of all components. A unit-

lengthvector is a vector whose length is one. A vector can be normalizedby dividing its

components with its length, converting the given vector to a unit-length one while keeping its

direction the same. For example, if a=, then the length of a, usually written as |a|, is

SQRT(50) and the normalized ais .

Inner Product

Given two vectors aand b, its inner product, written as a.b, is the sum of the products of

corresponding components. For example, if a= and b= , then a.b= 1*2 + 2*(-1) +

3*4 = 12.

The geometric meaning of the inner product of aand bis the following:

a.b= |a|.|b|cos(t)


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More precisely, the inner product of aand bis equal to the product of the length of a, the length of b,

and the cosine of the angle tbetween aand b. This is an important formula, because we know the

following facts about aand bonce the inner product becomes available:

1. If a.bis zero, where aand bare non-zero vectors, then cos(t) must be zero and, as a result, t

must be 90 degree. Therefore, aand bare perpendicular to each other.2. If a.bis equal to the product of lengths of aand b, the cosine of tis 1 and tis 0 degree. As a

result, aand bare parallel to each other and point to the same direction.

3. If a.bis equal to the negative product of lengths of aand b, the cosine of tis -1 and aand b

are parallel to each other but point to opposite directions.

Lines

A line is defined by a based point Band a direction vector dwhich gives the direction of the line.

Therefore, the vector equation of a line is

B+ td

where tis a parameter. In many applications, the direction vector is of unit-length.

Planes

A plane, in its vector form, is specified by a based point Band its normal vector n. For an arbitrary

point, or position vector, Xon the plane, the direction vector from the base point Bto X, X-B, must be

perpendicular to the normal vector n. Therefore, we have (X-B).nmust be zero. From (X-B).n=0, we

have the equation of a plane specified with a base point and its normal vector:

X.n- B.n= 0

Given the vector notation of lines and planes, it is very easy to compute the intersection point

of a line and a plane. Let the given line be A+td. Let the plane be defined with a base point B

and its normal vector n. Then, this plane has equation X.n=B.n. If the line intersects the

plane, there must be a value of tsuch that the corresponding point lies on the plane. That is,

there must be a tsuch that the point corresponding to this twould satisfy the plane equation.

Since a point on the line is A+td, plugging A+tdinto the plane equation yields

(A+td).n- B.n= 0

Rearranging the terms and solving for tyields

t= (B-A).n/ d.n

Therefore, plugging this tinto the line equation yields the intersection point.

In the above, if d.nis zero, tcannot be solved and consequently no intersection point exists.

The meaning of d.n= 0 is that dand nare perpendicular to each other. Since nis the normal

vector of a plane and dis perpendicular to n, dmust be parallel to the plane. If the line is

parallel to the plane, no intersection point exists.


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Cross Product

There is yet one more important concept about vector: the cross product of two vectors. Given two

vectors aand b, their cross product, written as a b, is defined as follows:

where a= < a1, a2, a3>, b= < b1, b2, b3>, and | |is a 2-by-2 determinant. In other words, the

cross product is

a b= < a2b3- a3b2, -(a1b3- a3b1), a1b2- a2b1>

The cross product of aand b, a b, is perpendicular to both aand band points to the direction basedon the right-handed system in the order of a, b, a b. Therefore, b apoints to the opposite direction

of a b. The length of a bis |a| |b| sin(t), where tis the acute angle between aand b. Hence, if a

and bare perpendicular to each other, the length of a bis simply |a| |b|.

Simple Curves and Surfaces

We have seen the simplest curves (lines) and surfaces (planes) in the previous page. Next tolines and planes, there are conics and quadric surfaces. Although conics and quadric surfaces

have been around for about 2000 years, they are still the most popular objects in many

computer aided design and modeling systems. We shall discuss conics, quadric surfaces and

tori on this page only. Consulting your calculus and/or geometry books should be very

helpful. Your linear algebra book should also cover some of these topics with a modern

approach.

The following figures show to you three different ways of cutting a cone with a plane. The

conic sections, from left to right, are an ellipse, a hyperbola and a parabola.


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Curves

Circles

The simplest non-linear curve is unquestionably the circle. A circle with center (a,b) and radius rhas

an equation as follows:

(x - a)2+ (x - b)

2= r

2

If the center is the origin, the above equation is simplified to

x2+y

2= r

2

The above equations are referred to as the implicitform of the circle. The parametric form of a circle

is

x= rcos(t)y= rsin(t)

The following is the parametric form of a circle whose center is not the origin:

x= a+ rcos(t)

y= b+ rsin(t)

The above parametric form uses trigonometric functions. We shall discuss a parametric form of a

circle without trigonometric functions later.

Conics in Normal Forms

A direct generalization of the circle is the so-called conic curvesor simply conics. Greeks knew about

conics very well. In fact, Apollonius of Perga (262 - 200 B.C.) wrote a book of several volumes about

conics. Conics are the intersection curves of a plane and a circular cone (i.e., a cone whose base is a

circle and whose axis is perpendicular to the base and through the center of the base circle).

There are three types of non-degenerate conics: ellipses, hyperbolas and parabolas. Ellipses

and hyperbolas are called centralconics because they have a center of symmetry, while

parabolas are non-central.

The normal form of an ellipse is the following implicit equation:

The axes of this ellipse are thex- andy-axis, aand bare the axis lengths, and the larger one

of aand bis the major axis while the smaller one is the minor axis. It is not difficult to see

that an ellipse in this form has the following parametric form:


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x= acos(t)

y= bsin(t)

The normal form of a hyperbola is the following implicit equation:

The definition of major axis and minor axis are identical to that of ellipses. The x-axis

intersects the curve at two points (a, 0) and (-a, 0) and, they-axis does not intersect the curve

at all. A possible parametric form of the above hyperbola is the following:

x= asec(t)

y= btan(t)

The center of an ellipse and hyperbola, in normal form, is the coordinate origin and the curve

is symmetric about its center and its axes.

The normal form of a parabola is the following implicit equation:

In this normal form, for any point (x,y) on a parabola, the value ofymust be positive and the

opening of this parabola is upward. The axis of this parabola is they-axis. It is intersecting to

note that the normal form of a parabola is already a parametric form. Or, if you like, you can

rewrite it into the following:

x= t

y= t2/ (4p)

Conics in General Form

Conics are degree two curves because their most general form is the following degree two implicit

polynomial:

In the above polynomial, the coefficients ofxy,xandyare 2B, 2Dand 2E, respectively. This

polynomial has six coefficients; however, dividing it with a non-zero coefficient would

reduce six to five. Thus, in general, five conditions can uniquely determine a conic. In linear

algebra, you perhaps have learned the way of reducing the above polynomial to a normalform using eigenvalues and eigenvectors.


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Frequently, we only want to know the curve type of a general second degree polynomial. In

this case, as long as the second degree equation represents a conic rather than two intersecting

or parallel lines, it can easily be done as follows:

1. IfB2A*C, the general equation represents a hyperbola.

ExpressionB2-A*Cis called the discriminantof the general second degree polynomial. Based on the

above, if the value of the discriminant is less than, equal to or greater than zero, the conic is an ellipse,

a parabola, or a hyperbola.

Conics in Matrix Form

One nice thing of conics is that its general form can be rewritten compactly using matrices. First, each

pointx= (x, y) is considered as a column vector whose third component is 1 and hence the transpose

is a row vector, written as xT= [ x, y, 1 ]. Next, the six coefficients of the general second degree

polynomial are used to construct a three-by-three symmetric matrix as follows:

It is not difficult to verify that the general second degree polynomial becomes

Now what you have learned from linear algebra can be applied to this matrix form.

Surfaces

Quadric Surfaces in Normal Forms

Quadric surfaces, or quadricsfor short, consist of the following different types: ellipsoids,

hyperboloids of one sheet, hyperboloids of two sheets, elliptic paraboloids, and hyperboloid

paraboloids. The following are their normal forms in implicit forms and their shapes:


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Ellipsoid

Hyperboloid of One Sheet

Hyperboloid of Two Sheets

Elliptic Paraboloid

Hyperbolic Paraboloid

These five quadric surfaces are normally referred to as rank four quadrics. There are two

types of rank three quadrics: cones and cylinders. Cylinders have three subtypes: ellipticcylinders, hyperbolic cylinders and parabolic cylinders as shown below:


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Cone

Elliptic Cylinder

Hyperbolic Cylinder

Parabolic Cylinder

Quadric Surfaces in General Form

The general form of quadric surfaces is the following:

It has ten coefficients; but, as mentioned in the discussion of conics, dividing the equation

with one of its non-zero coefficients reduces the number of coefficients to nine. Please also

note that except for the coefficients forx2

,y2

andz2

and the constant term, all coefficients hasa multiplier of 2.


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You might want to ask a question: is it possible to develop an algorithm for classifying the

quadrics just like we did for conics? More precisely, if a general second degree polynomial is

given, could we tell its type (i.e., ellipsoid, hyperboloid of one sheet, elliptic paraboloid, etc)

by simply looking at their coefficients? The answer is always a "yes"; but the computation

algorithm is quite complex. So, I would rather skip this algorithm. However, you can alwaysuse eigenvalues and eigenvectors to solve this problem.

Quadric Surfaces in Matrix Form

The equation of a general quadric can also be put into matrix form:

where (x,y,z) is the coordinates of a point. This form translates the general second

polynomial of a quadric to the following matrix form:

Note that it is exactly identical to that of a conic. Therefore, matrices help to bring conics and

quadrics into an identical form.

After knowing the matrix form of quadrics, we will be able to discuss the meaning of rank

four and rank three quadrics. Consider the symmetric matrix Qthat contains the coefficients

of a general second degree polynomial. The rank of a matrix is the number of non-zero

eigenvalues. Thus, rank four quadrics are those quadrics whose matrix Qare of rank four. It

is easy to see (from their normal forms) that ellipsoids, hyperboloids and paraboloids are rank

four quadrics, and cones and cylinders are rank three quadrics. If a general second degree

polynomial factors into the product of two distinct degree one polynomials (i.e., planes), Qwill have rank two.

Tori in Normal Form

A torus can be generated by rotating a circle, the minorcircle, about a line, the rotation axisor axis of

rotation. The moving circle has its center on another circle, the majorcircle. The radii of the major

and minor circles are referred to as the major radius and minor radius, denoted byRand r,

respectively.

If the rotation axis is thez-axis and the major circle lies on thexy-plane, the equation of thegenerated torus is the following:


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IfRis greater than r, the result is a commonly seen torus as shown in the middle of the

following figure.

IFRis equal to r, then all moving circles are tangent to the rotation axis at the coordinate

origin as shown in the right figure. IfRis less than r, all moving circles intersect the rotation

axis at two distinct points and the generated torus will have a olive like shape in the interior

of the torus as shown in the left figure.

Homogeneous Coordinates

One of the many purposes of using homogeneous coordinates is to capture the concept of

infinity. In the Euclidean coordinate system, infinity is something that does not exist.

Mathematicians have discovered that many geometric concepts and computations can be

greatly simplified if the concept of infinity is used. This will become very clear when we

move to curves and surfaces design. Without the use of homogeneous coordinates system, itwould be difficult to design certain classes of very useful curves and surfaces in computer

graphics and computer-aided design.

Let us consider two real numbers, aand w, and compute the value of a/w. Let us hold the

value of afixed and vary the value of w. As wgetting smaller, the value of a/wis getting

larger. If wapproaches zero, a/wapproaches to infinity! Thus, to capture the concept of

infinity, we use two numbers aand wto represent a value v, v=a/w. If wis not zero, the value

is exactly a/w. Otherwise, we identify the infinite value with (a,0). Therefore, the concept of

infinity can be represented with a number pair like (a, w) or as a quotient a/w.


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Let us apply this to thexy-coordinate plane. If we replacexandywithx/wandy/w, a function

f(x,y)=0 becomesf(x/w,y/w)=0. If functionf(x,y) = 0 is a polynomial, multiplying it with wn

will clear all denominators, where nis the degree of the polynomial.

For example, suppose we have a lineAx + By + C= 0. Replacingxandywithx/wandy/w

yieldsA(x/w) + B(y/w) + C= 0. Multiplying by wchanges it to

Ax + By + Cw = 0.

Let the given equation be a second degree polynomialAx2+ 2Bxy + Cy2+ 2Dx + 2Ey + F=

0. After replacingxandywithx/wandy/wand multiplying the result with w2, we have

Ax2+ 2Bxy + Cy2+ 2Dxw + 2Eyw + Fw2= 0

If you look at these two polynomials carefully, you will see that the degrees of all terms are

equal. In the case of a line, termsx,yand ware of degree one, while in the second degreepolynomial, all terms (i.e.,x2,xy,y2,xw,ywand w2) are of degree two.

Given a polynomial of degree n, after introducing w, all terms are of degree n. Consequently,

these polynomials are called homogeneouspolynomials and the coordinates (x,y,w) the

homogeneous coordinates.

Given a degree npolynomial in a homogeneous coordinate system, dividing the polynomial

with wnand replacingx/w,y/wwithxandy, respectively, will convert the polynomial back to

a conventional one. For example, if the given degree 3 homogeneous polynomial is the

following:

x3+ 3xy2- 5y2w+ 10w3= 0

the result is

x3+ 3xy2- 5y2+ 10 = 0

This works for three-dimension as well. One can replace a point (x,y,z) with (x/w,y/w,z/w)

and multiply the result by wraised to certain power. The resulting polynomial is a

homogeneous one. Converting a degree nhomogeneous polynomial inx,y,zand wback to

the conventional form is exactly identical to the two-variable case.

An Important Notes

Given a point (x,y,w) in homogeneous coordinates, what is its corresponding point in thexy-

plane? From what we discussed for converting a homogeneous polynomial back to its

conventional form, you might easily guess that the answer must be (x/w,y/w). This is correct.

Thus, a point (3,4,5) in homogeneous coordinates converts to point (3/5,4/5)=(0.6,0.8) in the


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xy-plane. Similarly, a point (x,y,z,w) in homogeneous coordinates converts to a point

(x/w,y/w,z/w) in space.

Conversely, what is the homogeneous coordinates of a point (x,y) in thexy-plane? It is simply

(x,y,1)! That is, let the wcomponent be 1. In fact, this is only part of the story, because the

answer is notunique. The homogeneous coordinates of a point (x,y) in thexy-plane is (xw,

yw, w) for any non-zero w. Why is this true? Because (xw,yw, w) is converted back to (x,y).

As a result, the following is important for you to memorize:

Converting from a homogeneous coordinates to a conventional one is unique; but,

converting a conventional coordinates to a homogeneous one is not.

For example, a point (4,2,3) in space is convert to (4w, 2w, 3w, w) for any non-zero w.

The Dimensionality of Homogeneous Coordinates

You perhaps have discovered that homogeneous coordinates need 3 and 4 components to

represent a point in thexy-plane and a point in space, respectively. Therefore, a point in space

(resp., thexy-plane) in homogeneous coordinates actually has four(resp., third) components.

Adding a fourth (resp., third) component whose value is 1 to the coordinates of a point in

space (resp., thexy-plane) converts it to its corresponding homogeneous coordinates.

Ideal Points or Points at Infinity

As mentioned at the very beginning of this page, homogeneous coordinates can easily capture

the concept of infinity. Let a point (x,y) be fixed and converted to a homogeneous coordinate

by multiplying with 1/w, (x/w,y/w,1/w). Let the value of wapproach to zero, then (x/w,y/w)

moves farther and farther away in the direction of (x,y). When wbecomes zero, (x/w,y/w)

moves to infinity. Therefore, we would say, the homogeneous coordinate (x,y,0) is the ideal

pointorpoint at infinityin the direction of (x,y).

Let us take a look at an example. Let (3,5) be a point in the xy-plane. Consider (3/w,5/w). If w

is not zero, this point lies on the liney= (5/3)x. Or, if you like the vector form, (3/w,5/w) is a

point on the line O+ (1/w)d, where the base point Ois the coordinate origin and dis the

direction vector . Therefore, as wapproaches zero, the point moves to infinity on the

line. This is why we say (x,y,0) is the ideal point or the point at infinity in the direction of

(x,y).

The story is the same for points in space, where (x,y,z,0) is the ideal point or point at infinity

in the direction of (x,y,z).

The concept of homogeneous coordinates and points at infinity in certain direction willbecome very important when we discuss representations of curves and surface.


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A Simple Geometric Interpretation

Given a homogeneous coordinate (x,y,w) of a point in thexy-plane, let us consider (x,y,w) to

be a point in space whose coordinate values arex,yand wfor thex-,y- and w- axes,

respectively. The line joining this point and the coordinate origin intersects the plane w= 1 at

a point (x/w,y/w, 1). Please verify this fact yourself. The following figure illustrates this

concept.

This transformation treats a two-dimensional homogeneous point as a point in three-

dimensional space and projects (from the coordinate origin) this three-dimensional point to

the plane w=1. Therefore, as a homogeneous point moves on a curve defined byhomogeneous polynomialf(x,y,w)=0, its corresponding point moves in three-dimensional

space, which, in turn, is projected to the plane w=1. Of course, (x/w,y/w) moves on a curve in

plane w=1.

The above figure also shows clearly that while the conversion from the conventional

Euclidean coordinates to homogeneous coordinates is unique, the opposite direction is not

because all points on the line joining the origin and (x,y,w) will be projected to (x/w,y/w,1).

This is also an important concept to be used in later lectures.

Geometric Transformations

When talking about geometric transformations, we have to be very careful about the object

being transformed. We have two alternatives, either the geometric objects are transformed or

the coordinate system is transformed. These two are very closely related; but, the formulae

that carry out the job are different. We only cover transforming geometric objects here.


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We shall start with the traditionalEuclidean transformationsthat do not change lengths and

angle measures, followed byaffine transformation.Finally, we shall talk aboutprojective

transformations.

Euclidean Transformations

The Euclidean transformations are the most commonly used transformations. An Euclidean

transformation is either a translation, a rotation, or a reflection. We shall discuss translations and

rotations only.

Translations and Rotations on the xy-Plane

We intend to translate a point in thexy-plane to a new place by adding a vector . It is not

difficult to see that between a point (x, y) and its new place (x', y'), we havex'=x+ handy'=y+ k.

Let us use a form similar to the homogeneous coordinates. That is, a point becomes a column vector

whose third component is 1. Thus, point (x,y) becomes the following:

Then, the relationship between (x, y) and (x', y') can be put into a matrix form like the following:

Therefore, if a line has an equationAx + By + C= 0, after plugging the formulae forxandy,

the line has a new equationAx' + By' + (-Ah - Bk + C) = 0.

If a point (x, y) is rotated an angle aabout the coordinate origin to become a new point (x', y'),

the relationships can be described as follows:

Thus, rotating a lineAx + By + C= 0 about the origin adegree brings it to a new equation:
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(Acosa-Bsina)x'+ (Asina+Bcosa)y'+ C= 0

Translations and rotations can be combined into a single equation like the following:

The above means that rotates the point (x,y) an angle aabout the coordinate origin and

translates the rotated result in the direction of (h,k). However, if translation (h,k) is applied

first followed by a rotation of angle a(about the coordinate origin), we will have the

following:

Therefore, rotation and translation are not commutative!

In the above discussion, we always present two matrices, Aand B, one for transforming xto

x'(i.e., x'=Ax) and the other for transforming x'to x(i.e., x=Bx'). You can verify that the

product of Aand Bis the identity matrix. In other words, Aand Bare inverse matrices of

each other. Therefore, if we know one of them, the other is the inverse of the given one. For

example, if you know Athat transforms xto x', the matrix that transforms x'back to xis the

inverse of A.

Let Rbe a transformation matrix sending x'to x: x=Rx'. Plugging this equation of xinto a

conic equation gives the following:


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Rearranging terms yields

This is the new equation of the given conic after the specified transformation. Note that the

new 3-by-3 symmetric matrix that represents the conic in a new position is the following:

Now you see the power of matrices in describing the concept of transformation.

Translations and Rotations in Space

Translations in space is similar to the plane version:

The above translates points by adding a vector .

Rotations in space are more complex, because we can either rotate about the x-axis, they-axis

or thez-axis. When rotating about thez-axis, only coordinates ofxandywill change and the

z-coordinate will be the same. In effect, it is exactly a rotation about the origin in thexy-

plane. Therefore, the rotation equation is

With this set of equations, letting abe 90 degree rotates (1,0,0) to (0,1,0) and (0,1,0) to (-

1,0,0). Therefore, thex-axis rotates to they-axis and they-axis rotates to the negative

direction of the originalx-axis. This is the effect of rotating about thez-axis 90 degree.


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Based on the same idea, rotating about thex-axis an angle ais the following:

Let us verify the above again with abeing 90 degree. This rotates (0,1,0) to (0,0,1) and

(0,0,1) to (0,-1,0). Thus, they-axis rotates to thez-axis and thez-axis rotates to the negative

direction of the originaly-axis.

But, rotating about they-axis is different! It is because the way of measuring angles. In a

right-handed system, if your right hand holds a coordinate axis with your thumb pointing in

the positive direction, your other four fingers give the positive direction of angle measuring.

More precisely, the positive direction for measuring angles is from the z-axis tox-axis.

However, traditionally the angle measure is from thex-axis to thez-axis. As a result, rotating

an angle aabout they-axis in the sense of a right-handed system is equivalent to rotating an

angle -ameasuring from thex-axis to thez-axis. Therefore, the rotation equations are

Let us verify the above with rotating about they-axis 90 degree. This rotates (1,0,0) to (0,0,-

1) and (0,0,1) to (1,0,0). Therefore, thex-axis rotates to the negative direction of thez-axis

and thez-axis rotates to the originalx-axis.

A rotation matrix and a translation matrix can be combined into a single matrix as follows,

where the r's in the upper-left 3-by-3 matrix form a rotation andp, qand rform a translation

vector. This matrix represents rotations followed by a translation.


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You can apply this transformation to a plane and a quadric surface just as what we did for

lines and conics earlier.

Affine TransformationsEuclidean transformations preserve length and angle measure. Moreover, the shape of a geometric

object will not change. That is, lines transform to lines, planes transform to planes, circles transform

to circles, and ellipsoids transform to ellipsoids. Only the position and orientation of the object will

change. Affine transformations are generalizations of Euclidean transformations. Under affine

transformations, lines transforms to lines; but, circles become ellipses. Length and angle are not

preserved. In this section, we shall discuss scaling, shear and general affine transformations.

Scaling

Scaling transformations stretch or shrink a given object and, as a result, change lengths and angles.So, scaling is not an Euclidean transformation. The meaning of scaling is making the new scale of a

coordinate directionptimes larger. In other words, thexcoordinate is "enlarged"ptimes. This

requirement satisfiesx'=p xand thereforex=x'/p.

Scaling can be applied to all axes, each with a differentscaling factor. For example, if thex-,

y- andz-axis are scaled with scaling factorsp, qand r, respectively, the transformation matrix

is:

Shear

The effect of a shear transformation looks like ``pushing'' a geometric object in a direction parallel to

a coordinate plane (3D) or a coordinate axis (2D). In the following, the red cylinder is the result ofapplying a shear transformation to the yellow cylinder:

How far a direction is pushed is determined by ashearing factor. On thexy-plane, one can

push in thex-direction, positive or negative, and keep they-direction unchanged. Or, one can


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push in they-direction and keep thex-direction fixed. The following is a shear transformation

in thex-direction with shearing factor a:

The shear transformation in they-direction with shearing factor bis the following:

In space, one can push in two coordinate axis directions and keep the third one fixed. The

following is the shear transformation in bothx- andy-directions with shearing factors aand b,

respectively, keeping thez-coordinate the same:

Let us take a look at the effect of this shear transformation. Expanding the matrix equation

gives the following:

x'=x+ az

y'=y+ bz

z'=z

Thus, a point (x, y, z) in space is transformed to (x+ az,y+ bz,z). Therefore, thez-coordinate does

not change, while (x, y) is ``pushed'' in the direction of (a, b, 0) with a factorz.

The following is the shear transformation inxz-direction:


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The following is the shear transformation inyz-direction:

General Affine Transformations

The general affine transformation matrix has the following form:

Comparing with all previous discussed matrices, rotations and translations included, you will

see that all of them fit into this form and hence are affine transformations. Affine

transformations do not alter the degree of a polynomial, parallel lines/planes are transformed

to parallel lines/planes, and intersecting lines/plane are transformed to intersecting lines and

planes. However, affine transformations do not preserve lengths and angle measures and as a

result they will change the shape of a geometric object. The following shows the result of a

affine transformation applied to a torus. A torus is described by a degree four polynomial.

The red surface is still of degree four; but, its shape is changed by an affine transformation.


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Note that the matrix form of an affine transformation is a 4-by-4 matrix with the fourth row 0,

0, 0 and 1. Moreover, if the inverse of an affine transformation exists, this affine

transformation is referred to as non-singular; otherwise, it issingular. We do not use singular

affine transformations in this course.

Projective Transformations

Projective transformations are the most general "linear" transformations and require the use of

homogeneous coordinates. Given a point in space in homogeneous coordinate (x,y,z,w) and its image

under a projective transform (x',y',z',w'), a projective transform has the following form:

In the above, the 4-by-4 matrices must be non-singular (i.e., invertible). Therefore, projective

transformations are more general than affine transformations because the fourth row does not

have to contain 0, 0, 0 and 1.

Projective transformation can bring finite points to infinity and points at infinity to finite

range. Let us take a look at an example. Consider the following projective transformation:

Obviously, this transformation sends (x,y,w)=(1,0,1) to (x',y',w') = (1,-1,0). That is, this

projective transformation sends (1,0) on thexy-plane to the point at infinity in direction . From the right-hand side of the matrix equation x=Px'we have


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x= 2x' + y'

y=x' + y'

w= 2x' + y' + w'

Let us consider a circlex^2 +y^2 = 1. Plugging the above equations into the circle equation changes it

to the following:

x2+ 2xy+y

2- 4xw- 2yw- w

2= 0

Dividing the above by w^2 to convert it back to conventional form yields

x2+ 2xy+y

2- 4x- 2y- 1 = 0

This is a parabola! (Why?) Therefore, a circle that has no point at infinity is transformed to a parabola

that does have point at infinity.

While projective transformations, like affine transformations, do not change the degree of a

polynomial, two parallel (i.e., intersecting) lines/planes can be transformed to two

intersecting (i.e., parallel) lines/planes. Please verify this fact yourself.

Although we do not use these facts and the concept of projective transformations

immediately, it will be very helpful in later lectures.

Matrix Multiplication and Transformations

We have introduced to you several transformations. We always show to you two forms, one from xto

x'and the other the inverse from x'to x. In many cases, one may need several transformations to

bring an object to its desired position. For example, one may need a transformation in matrix form

q=Apbringing pto q, followed by a second transformation r=Bqbringing qto r, followed by yet

another transformation s=Crbringing rto s. The net effect of p-> q-> r-> scan be summarized into

a single transformation represented by the product of all involved matrices. Note that the first (resp.,

last) transformation matrix is the right-most (resp., left-most) in the multiplication sequence.

s= Cr= C(Bq) = CBq= CB(Ap) = CBAp

Therefore, to compute the net effect, we just compute CBAand use it as a single transformation,

which brings pto s.

Let us take a look at an example. We want to perform the following transformations to an

object:

1. Scale in thex-direction using a scale factor 5 (i.e., making it five times larger).

2. Followed by a rotation aboutz-axis 30 degree

3. Followed by a shear transformation inx- andy-direction with shearing factor 2 and 3,

respectively.

4. Followed by a transformation moving the point in the direction of < 2, 1, 2 >.

Let the scaling, rotation, shearing and translation matrices be A, B, Cand D, respectively. With

previous discussion, we have the following, where matrix H= DCBAis the net effect:


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Therefore, the net effect of transforming a point xof the initial object to the corresponding

point x'after the above four transformations is computed as x'= Hx= DCBAx.

Problems

1. Given a line B+tdand a plane with base point Aand normal vector n, what is the

condition for the line is perpendicular to the plane? What is the condition for the line

to be parallel to the plane?

References

Most of this week's material are from Bowyer and Wookwark's little book. The example

showing the bad effect of the associative law is from Colonna's paper. Finally, you can

find more about floating pointing computation problems is Acton's book.

Wolfgang Boehm and Hartmut Prautzsch, Geometr ic Concepts for Geometr icDesign, AK Peters, Wellesley, MA, 1994.


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Michael E. Mortenson, Computer Graphics: An I ntr oduction to the Mathematics

and Geometry, Heinemann Newnes, Oxford, UK, 1989.

3. cad dr.c.k.shene geometric concepts

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