Solution for Computer Graphics and Virtual Reality · 2020-01-28 · Solution for Computer Graphics and Virtual Reality May -2016 ... With respect to 3D transformation ,describe the

Mumbai University I.T (Semester V)

1 |www.brainheaters.in

Solution for Computer Graphics

and Virtual Reality

May -2016

Index Q.1)

a) ………………………………………………………………………….2

b) ………………………………………………………………………….2

c) ………………………………………………………………………….3

d) ………………………………………………………………………….3-5

Q.2)

a) ………………………………………………………………………….5-7

b) ………………………………………………………………………….7-9

Q.3)

a) ………………………………………………………………………….10-12

b) ………………………………………………………………………….N.A

Q.4)

a) ………………………………………………………………………….12-16

b) ………………………………………………………………………….16-17

Q.5)

a) ………………………………………………………………………….N.A

b) ………………………………………………………………………….18-19

Q.6)

a) ………………………………………………………………………….19-20

b) ………………………………………………………………………….20-22

c) ………………………………………………………………………….22-24

d) ………………………………………………………………………….24-26

e) …………………………………………………………………………26-27



Q1)

(a) Differentiate between Raster scan display and Random scan display

Ans:

(b) Prove that two successive rotation transformation are additive.

Ans:

About a line



(c) Show that the transformation matrix for the reflection a line y=x is

equivalent to reflection to x-axis followed by counter clockwise rotation of

900

Ans:

(d) Explain 3D trackers and enumerate some important trackers characteristics.

Ans:



Characteristics



Q2)

(a) Specify and highlights and drawbacks of Bezier curve .Construct the Bezier

curve of order three with control points P1(0,0) , P2(1,3) P3(4,2) and

P4(2,1) .Generate at least five points on the curve.

Ans:





(b) Write DDA line drawing Algorithm .Compare DDA with Bresenham’s Line

drawing Algorithm.Calculate the pixel co-ordinates of line AB using DDA

algorithm where A=(0,0) and B =(4,6).

Ans:







Q3)

(a) Let ABCD be the rectangular window with A (20,20) ,B(90,20) C(90,70)

And D( 20,70) .Find the region codes for endpoints and use cohen Sutherland

algorithm to clip the lines P1 P2 with P1 (10,30) and P2(80,90)

Ans:





(b) With respect to 3D transformation ,describe the steps to be carried out when

an object is to be rotated about an arbitrary axis. Specify all the required

matrices .State your assumptions clearly.

Ans: N.A

Q4)

(a) Explain the flood fill algorithm for 4 connected and 8 connected .What are

its advantages over boundary fill algorithm.

Ans:

Flood Fill Algorithm

Sometimes we come across an object where we want to fill the area and its boundary

with different colors. We can paint such objects with a specified interior color instead

of searching for particular boundary color as in boundary filling algorithm.

Instead of relying on the boundary of the object, it relies on the fill color. In other

words, it replaces the interior color of the object with the fill color. When no more

pixels of the original interior color exist, the algorithm is completed.



Once again, this algorithm relies on the Four-connect or Eight-connect method of

filling in the pixels. But instead of looking for the boundary color, it is looking for all

adjacent pixels that are a part of the interior.

4-Connected & 8-ConnectedPolygon

In this technique 4-connected pixels are used as shown in the figure1. We are putting

the pixels above, below, to the right, and to the left side of the current pixels and this

process will continue until we find a boundary with different color.

Algorithm

Step 1 − Initialize the value of seed point (seedx, seedy), fcolor and dcol.

Step 2 − Define the boundary values of the polygon.

Step 3 − Check if the current seed point is of default color, then repeat the steps 4 and

5 till the boundary pixels reached.

If getpixel(x, y) = dcol then repeat step 4 and 5



Step 4 − Change the default color with the fill color at the seed point.

setPixel(seedx, seedy, fcol)

Step 5 − Recursively follow the procedure with four neighborhood points.

FloodFill (seedx – 1, seedy, fcol, dcol)

FloodFill (seedx + 1, seedy, fcol, dcol)

FloodFill (seedx, seedy - 1, fcol, dcol)

FloodFill (seedx – 1, seedy + 1, fcol, dcol)

Step 6 − Exit

There is a problem with this technique. Consider the case as shown below in figure 3

where we tried to fill the entire region. Here, the image is filled only partially. In such

cases, 4-connected pixels technique cannot be used.

8-Connected Polygon

In this technique 8-connected pixels are used as shown in the figure2. We are putting

pixels above, below, right and left side of the current pixels as we were doing in 4-

connected technique.

In addition to this, we are also putting pixels in diagonals so that entire area of the

current pixel is covered. This process will continue until we find a boundary with

different color.

Algorithm

Step 1 − Initialize the value of seed point (seedx, seedy), fcolor and dcol.

Step 2 − Define the boundary values of the polygon.

Step 3 − Check if the current seed point is of default color then repeat the steps 4 and

5 till the boundary pixels reached

If getpixel(x,y) = dcol then repeat step 4 and 5

Step 4 − Change the default color with the fill color at the seed point.

setPixel(seedx, seedy, fcol)

Step 5 − Recursively follow the procedure with four neighbourhood points



FloodFill (seedx – 1, seedy, fcol, dcol)

FloodFill (seedx + 1, seedy, fcol, dcol)

FloodFill (seedx, seedy - 1, fcol, dcol)

FloodFill (seedx, seedy + 1, fcol, dcol)

FloodFill (seedx – 1, seedy + 1, fcol, dcol)

FloodFill (seedx + 1, seedy + 1, fcol, dcol)

FloodFill (seedx + 1, seedy - 1, fcol, dcol)

FloodFill (seedx – 1, seedy - 1, fcol, dcol)

Step 6 − Exit

The 4-connected pixel technique failed to fill the area as marked in the figure 3 which

won’t happen with the 8-connected technique.

Comparison



(b) Explain an algorithm which uses parametric equation of line clipping .Using

same algorithm find the line segment A(10,10) and B(70,40) after it is

clipped against the window of two vertices (20,10) and (40,50).

Ans: The solution to the sum of this question is not avail available. Answer to

the theory part is given below

Liang-Barsky Line Clipping

The ideas for clipping line of Liang-Barsky and Cyrus-Beck are the same. The only

difference is Liang-Barsky algorithm has been optimized for an upright rectangular

clip window. So we will study only the idea of Liang-Barsky.

Liang and Barsky have created an algorithm that uses floating-point arithmetic but

finds the appropriate end points with at most four computations. This algorithm uses

the parametric equations for a line and solves four inequalities to find the range of the

parameter for which the line is in the viewport.



Let be the line which we want to study. The parametric

equation of the line segment from gives x-values and y-values for every point in

terms of a parameter tthat ranges from 0 to 1. The equations are

and

We can see that when t = 0, the point computed is P(x1,y1); and when t = 1, the point

computed is Q(x2,y2).

Algorithm

1. Set and

2. Calculate the values of tL, tR, tT, and tB (tvalues).

if or ignore it and go to the next edge

otherwise classify the tvalue as entering or exiting value (using inner

product to classify)

if t is entering value set ; if t is exiting value set

3. If then draw a line from (x1 + dx*tmin, y1 + dy*tmin) to (x1 +

dx*tmax, y1 + dy*tmax)

4. If the line crosses over the window, you will see (x1 + dx*tmin, y1 + dy*tmin)

and (x1 + dx*tmax, y1 + dy*tmax) are intersection between line and edge.

https://www.cs.helsinki.fi/group/goa/viewing/leikkaus/tvalue.html

https://www.cs.helsinki.fi/group/goa/viewing/leikkaus/enter.html

https://www.cs.helsinki.fi/group/goa/viewing/leikkaus/intersec.html



Q5)

(a) Consider a triangle ABC whose coordinates are A(10,20) B(30,40) and

8C(50,20) .Perform the following transformation (specify the matrices that

are used)

(i) Translate the given triangle by 3 units in X direction and -2 unit in Y

direction .

(ii) Rotate the given triangle by 30

(iii) Reflect the given triangle about X=Y.

(iv) Scale the given triangle uniformly by 2 units.

Ans: N.A

(b) What is significance of modelling in virtual reality? Explain any modelling

technique used in virtual reality).

Ans:

Real-Life Situations

Virtual reality modeling is used to create interactive scenarios which reflect real-life

situations. Virtual reality e-learning can be used to simulate the way equipment

responds; emulate the way machinery works or to replicate soft skills such as human

actions and behaviour. Anything is possible; just let your imagination go wild!

Making Learning Fun and Interesting

As a child you watched, listened and played. You were intrigued by colours, shapes

and sounds and the only way you found out how something worked was by playing

and using it. Virtual reality simulations use these basic principles of learning to

produce fun, compelling and memorable end results. We engage the user with

breathtaking graphics, informative audio and interactive scenarios using our 3D

virtual environments to give a sense of really being there.

Save on Cost

Do you have limited training resources, equipment that is scarce or expensive, too

many people to train at one time? If some of these limitations apply to you then using

virtual reality could be a perfect solution for your needs.



By modelling your equipment, possibly down to the last detail you could distribute a

training programme to all your employees or learners that will allow them to interact

with it, follow best practice procedures or carry out fault finding scenarios, all

without having to access (and possibly damage) the real item.

Complex Situations Made Easy

Complicated pieces of equipment, processes or systems can be recreated using a

number of techniques. This form of e-learning allows users to learn about

mechanisms and processes that would be physically or logistically difficult to do so

in other conditions.

A Safe Environment

There are some occasions when training your staff is expensive and it can sometimes

be hazardous, with real risk of personal injury or damage to expensive equipment.

One can develop highly interactive, virtual reality environments to help with these

issues. By creating an environment which simulates a potentially harmful real-life

situation or replicating a piece of dangerous equipment, Sinteractive scenarios

remove these concerns and help the user gain a knowledge and understanding of the

subject matter without being put into a costly or harmful environment.

Q6)

(a) Homogeneous Coordinates

Ans:

In general, when you want to perform a complex transformation, you usually make it

by combining a number of basic transformations. The above equation for q, however,

is awkward to read because scaling is done by matrix multiplication and translation is

done by vector addition. In order to represent all transformations in the same form,

computer scientists have devised what are called homogeneous coordinates. Do not

try to apply any exotic interpretation to them. They are simply a mathematical trick to

make the representation be more consistent and easier to use.

Homogeneous coordinates (HC) add an extra virtual dimension. Thus 2D HC are

actually 3D and 3D HC are 4D. Consider a 2D point p = (x,y). In HC, we represent p

as p = (x,y,1). An extra coordinate is added whose value is always 1. This may seem

odd but it allows us to now represent translations as matrix multiplication instead of



as vector addition. A translation (dx, dy) which would normally be performed as q =

(x,y) + (dx, dy) now is written as

q = T p = =

Now, we can write the scaling about a fixed point as simply a matrix multiplication:

q = (-T) S T p = A p,

where A = (-T) S T

The matrix A can be calculated once and then applied to all the points in the object.

This is much more efficient than our previous representation. It is also easier to

identify the transformations and their order when everything is in the form of matrix

multiplication.

The matrix for scaling in HC is

S =

and the matrix for rotation is

R =

(b) Text Clipping

Ans:

Various techniques are used to provide text clipping in a computer graphics. It

depends on the methods used to generate characters and the requirements of a

particular application. There are three methods for text clipping which are listed

below −

All or none string clipping

All or none character clipping



Text clipping

The following figure shows all or none string clipping −

In all or none string clipping method, either we keep the entire string or we reject

entire string based on the clipping window. As shown in the above figure, STRING2

is entirely inside the clipping window so we keep it and STRING1 being only

partially inside the window, we reject.

The following figure shows all or none character clipping −

This clipping method is based on characters rather than entire string. In this method if

the string is entirely inside the clipping window, then we keep it. If it is partially

outside the window, then −

You reject only the portion of the string being outside

If the character is on the boundary of the clipping window, then we discard that

entire character and keep the rest string.

The following figure shows text clipping −



This clipping method is based on characters rather than the entire string. In this

method if the string is entirely inside the clipping window, then we keep it. If it is

partially outside the window, then

You reject only the portion of string being outside.

If the character is on the boundary of the clipping window, then we discard

only that portion of character that is outside of the clipping window.

(c) Fractals

Ans:

Fractals are very complex pictures generated by a computer from a single formula.

They are created using iterations. This means one formula is repeated with slightly

different values over and over again, taking into account the results from the previous

iteration.

Fractals are used in many areas such as −

Astronomy − For analyzing galaxies, rings of Saturn, etc.

Biology/Chemistry − For depicting bacteria cultures, Chemical reactions,

human anatomy, molecules, plants,

Others − For depicting clouds, coastline and borderlines, data compression,

diffusion, economy, fractal art, fractal music, landscapes, special effect, etc.

Generation of Fractals

Fractals can be generated by repeating the same shape over and over again as shown

in the following figure. In figure (a) shows an equilateral triangle. In figure (b), we



can see that the triangle is repeated to create a star-like shape. In figure (c), we can

see that the star shape in figure (b) is repeated again and again to create a new shape.

We can do unlimited number of iteration to create a desired shape. In programming

terms, recursion is used to create such shapes.

Geometric Fractals

Geometric fractals deal with shapes found in nature that have non-integer or fractal

dimensions. To geometrically construct a deterministic (nonrandom) self-similar

fractal, we start with a given geometric shape, called the initiator. Subparts of the

initiator are then replaced with a pattern, called the generator.

As an example, if we use the initiator and generator shown in the above figure, we

can construct good pattern by repeating it. Each straight-line segment in the initiator

is replaced with four equal-length line segments at each step. The scaling factor is

1/3, so the fractal dimension is D = ln 4/ln 3 ≈ 1.2619.

Also, the length of each line segment in the initiator increases by a factor of 4/3 at

each step, so that the length of the fractal curve tends to infinity as more detail is

added to the curve as shown in the following figure −



(d) B-Spline Curves

Ans:

The Bezier-curve produced by the Bernstein basis function has limited flexibility.

First, the number of specified polygon vertices fixes the order of the resulting

polynomial which defines the curve.

The second limiting characteristic is that the value of the blending function is

nonzero for all parameter values over the entire curve.

The B-spline basis contains the Bernstein basis as the special case. The B-spline basis

is non-global.

A B-spline curve is defined as a linear combination of control points Pi and B-spline

basis function Ni,

k (t) given by

C(t)=∑ni=0PiNi,k(t),

n≥k−1, tϵ[tk−1,tn+1]



Where,

{pi

: i=0, 1, 2….n} are the control points

k is the order of the polynomial segments of the B-spline curve. Order k means

that the curve is made up of piecewise polynomial segments of degree k - 1,

the Ni,k(t)

are the “normalized B-spline blending functions”. They are described by the

order k and by a non-decreasing sequence of real numbers normally called the

“knot sequence”.

ti:i=0,...n+K

Properties of B-spline Curve

B-spline curves have the following properties −

The sum of the B-spline basis functions for any parameter value is 1.

Each basis function is positive or zero for all parameter values.

Each basis function has precisely one maximum value, except for k=1.

The maximum order of the curve is equal to the number of vertices of defining

polygon.

The degree of B-spline polynomial is independent on the number of vertices of

defining polygon.

B-spline allows the local control over the curve surface because each vertex

affects the shape of a curve only over a range of parameter values where its

associated basis function is nonzero.

The curve exhibits the variation diminishing property.

The curve generally follows the shape of defining polygon.

Any affine transformation can be applied to the curve by applying it to the

vertices of defining polygon.

The curve line within the convex hull of its defining polygon.



(e) Morphing and wraping

Ans:

Morphing

Morphing is a familiar technology to produce special effects in image or videos.

Morphing is common in entertainment industry. Morphing is widely used in movies,

animation games etc. In addition to the usage of entertainment industry, morphing

can be used in computer based trainings, electronic book illustrations, presentations,

education purposes etc. morphing software is widely available in internet.

Animation industry looking for advanced technology to produce special effects on

their movies. Increasing customers of animation industry does not satisfy with the

movies with simple animation. Here comes the significance of morphing.

The Word "Morphing" comes from the word "metamorphosis" which means change

shape, appearance or form. Morphing is done by coupling image warping with colour

interpolation. Morphing is the process in which the source image is gradually



distorted and vanished while producing the target image. So earlier images in the

sequence are similar to source image and last images are similar to target image.

Middle image of the sequence is the average of the source image and the target

image.

Morphing techniques may be classified into two based on the ways to specify their

features. They are mesh based methods and feature based methods. In mesh based

methods, features of the image specified by a non uniform mesh. Feature based

methods, features of the image specifies as line segment or a set of points. Feature

based methods are popular.

Warping:

A warp is a 2-D geometric transformation and generates a distorted image when it is applied to an image. Warping an image means : apply a given deformation to it. Two ways to warp an image:- Some pixels in the destination image may not be mapped. Each pixel in source image is mapped to an appropriate pixel in destination image.

This method goes through each pixel in the destination image and samples an appropriate source image pixel. All destination image pixels are mapped to some source image pixel. This mapping is used in the Beier/Neely line morphing method.

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Solution for Computer Graphics and Virtual Reality · 2020-01-28 · Solution for Computer Graphics and Virtual Reality May -2016 ... With respect to 3D transformation ,describe the