An Exchange Of e-mails With Glenda Ramilo

The letters below were in connection with the solicitation sent by the CEA student body thru their adviser, Engr. Glenda Ramilo-Cabauatan. It is an acknowlegement of the small amount we were able to raise from Louisians in Dubai. I am posting the same to thank and give credit to those who responded to the appeal.

.

Dear Sir,

I don't know if you received my text message last Saturday. I sent three messages and I hope you were able to get them. Anyways, I have received the money that you and Carlo sent (P5,114.71 and P7,224.08) ' documentary stamp tax deducted already'. for a total of P12,338.79. Thanks to all the contributors and we are still hoping that more will come soon. I have already informed Arrianne about it and told him to send you an email to acknowledge all those who have contributed. The opening of intramurals is scheduled this afternoon, but the cheering competition will be on the 11th of September. We are all hoping here that before the day of the competition comes, we have already purchased the much needed drum set. Again, thank you very much for your kindness! Please extend our gratitude to everyone!

God bless!

Respectfully yours,

Engr. Glenda Ramilo-Cabauatan

<<<<<<<<<<<<< >>>>>>>>>>>>>>

Dear Glenda,

In behalf of our Louisian colleauges here in the UAE, I am pleased to know that you have received the small amount that we were able to raise. I have tried my best to get in touch with everyone here but I was able to reach only a few and even fewer of them were willing to contribute. It is understandable considering the climate of financial uncertainty and job insecurity now gripping all of us. Nonetheless, there are always people who are willing to help despite their own personal circumstances and I think we need to mention and acknowledge them individually.

From Dubai-----------------------------------------------AED 400.00
Engr. Condrado Noble
Engr. Theody Raul Nones
Engr. Isagani Espejo

From Sharjah-------------------------------------------AED 250.00
Engr. Flordeliza Collado-Doctolero
Engr. Joel Lasquite
Engr. Jeffrey Jaramillo

From Abu Dhabi---------------------------------------AED 350.00
Engr. Carlo Chan
Engr. Ed Blancas
Engr. Ludy Aquino
Engr. Jose Gapasin
T O T A L -----------------------------------------------P12,338.79 (after fees and taxes)

We hope you can use this amount, however limited it is, to a worthy purpose.

Thank you and best regards to everyone.


Engr. Isagani Espejo
for the Louisians Engineers
in the UAE

Yellow Fever: A Cleansing Plague

by icarus
.

Dejavu. There is again an outbreak of yellow fever in Manila and the rest of the country. This most virulent virus has afflicted millions and millions of Filipinos across the archipelago. And it is more than welcome. Just like the old days that culminated to EDSA '86. People by the hundreds of thousands pour out into the streets to have one last glimpse of Tita Cory in gloriuos garbs of yellow. The metropolis is deluged with yellow banners, yellow headbands, yellow t-shirts - people wearing yellow something and flashing the "L" sign amidst rains of yellow confetti. The veterans of People power, long retired and given up on the eroded promise of EDSA had awakened from their lethargy.
.
It took Ninoy's death to awaken the nation's wrath against the excesses of the dictatorship. Now it looks like Cory's death would have the same effect for this corrupt regime propped by cheating, lying, and deceit. For why are people leaving their comfort zones and braving the heat and the rain just to see the funeral convoy pass by? Why are they chanting “Cory! Cory!” for somebody who can’t hear it anymore? Why are they flashing the defiant Laban sign? Surely it’s not for the beloved Cory to see. Consciously or subconsciously, this public display of love for Cory is also a subtle expression of the people’s repressed hatred and derision for the pa-cute occupant of Malacanang. For inevitably there will be comparisons that could only emphasize how immaculate Cory was and how shameless Gloria is; how great and humble the ex-president was and how deceitful and arrogant the current one is. And the people would know what they missed and what they were being denied; of what should be and what should be not.
.
Take a cue from the politicians who have extra sharpened senses for these cryptic messages and body language that hints on the pulse of the people. That’s the reason why they come a-flocking to associate themselves with the Aquino's and distance themselves from the Arroyo's. Like rats jumping out of a sinking ship. But any one with an open mind could read the writings on the wall. Verily, verily I tell you - one of these days, the people’s fury from within will manifest itself unmistakably to the Administration, either in a tumultuous ousting or a whipping at the polls.

.

The days of these arrogant leaders are numbered; they will be disseminated by the cleansing plague called yellow fever! Lets keep the infection rampant and unchecked. Spread the Virus.

.
Mabuhay ang Pilipinas!
.
.
.

An Opportunity To Help

Dear Louisian Engineers,

Earlier today, I was pleasantly surprised by an e-mail from our fellow Louisian and board topnotcher Glenda Ramilo-Cabauatan, a former student of mine who is now one of the pillars of the Civil Engineering Faculty at Saint Louis College. Currently, she is the Adviser for the CEA Student Council headed by the Gov. Arrianne Fernandez. Attached on her e-mail and reproduced here in full is a solicitation letter from the CEA and whose content is self-explanatory.

.

Fellow Louisians in the UAE, please allow me to rally our group and once more appeal for your usual generous donations to support the worthy endeavors of our very own College of Engineering and Architecture. Not everyone, and certainly not so many, are in a position to help like you can - like a mohandesh can. So let us take this as a rare opportunity to show once more that the products of the College of Engineering are the best – doing well and ready to share their blessings – then, now and in the foreseeable future.
.
Due to the very short notice that we are given, there is not much time left to gather as one to pledge our support. (We can talk and plan about a get-together later). Thus, I have requested the following persons below to receive and transmit your donations directly to the CEA account to be provided by Glenda later. You can also send your donations directly to the same account but the cost of remittance would substantially eat into your donated amount. Rest assured that your donations will be acknowledged individually and transparently on this blogsite.

.
Engr. Norilyn Castillo (050-2674593)
Engr. Michelle Mayo (050-5031498)
Engr. Angelo Canedo (050-7197418)
Engr. Carlo Chan (050-2857836)
Engr. Ritchie Flores (050-9750141)
and yours truly (050-3543906)
.
I have also requested Glenda and Arrianne to identify more worthwhile projects other than the Intramurals that our group could support for this school year.

.

Thank you and best regards to everyone.
.

.

Very truly yours,

Engr. Isagani S. Espejo

Louisian 260162

.

.

.

double-click image to enlarge:

.

IMHO

SONA babitz!

by icarus

.

.

The SONA (State Of The Nation Address) of GMA yesterday was a bummer to put it gently. It’s the same package of bogus claims, tall tales, self-serving propagandas, excuses, and uncalled for swipes at her political rivals. If we have to believe the President, the country’s economic performance is on all-time high, jobs generation and employment opportunities are phenomenally great, graft and corruption are non-existent, human rights violations and involuntary disappearances are just pigments of her detractor's imagination; and therefore she is the messiah the Philippines had been waiting for.

.

Now, you don’t have to have a PhD to see through all of these. In fact, it is the people in the lowest rungs of society who could tell most graphically how the economy has stagnated at rock-bottom during her watch. It is the ever-increasing number of the unemployed and the under-employed which could prove most eloquently the lack or non-existence of jobs. So how did these Doctors of Economics at the NEDA come up with the rosy employment figures? Its easy, I tell you. They simply re-classified jobs and changed the definitions. For example, if bum Juan could not find a salaried job but drives her mom to market regularly on weekends then by their definition and in their statistics, Juan is employed. Congratulations, Juan! Be grateful that you have a job. Now, why are you asking for salary, you ingrate!

.

Is there still graft and corruption? Is there still sand in the Sahara? It’s the most self-evident fact and yet she refuses to see. Isn’t it obvious when they feed the millions of undernourished school children with P5.00 pack instant noodles and claim for P18.00 disbursement from the government budget? For a million school-children that’s 13M every school day! You don’t call that a charitable Feeding Program, do you? It’s a heartless stealing from the bowl of poor children and giving it to the sharks – Oh, it’s a Feeding Program, alright! How callous could we get. Are you people enraged yet? That’s just the mildest case.

.

How about Human Rights? Well, have you come across the names Karen Empeno, Sherlyn Cadapan, Melissa Roxas, Jonas Burgos? Google their cases and seethe in righteous anger! Be shocked at the brazenness and impunity by which the all-powerful authorities determine who lives and who dies. Shudder at the thought that they could come and drag you or your loved one from sleep and turned into grim statistics. While everyone watch and cower in helplessness.

.

Watching that speech made me really furious. Her smile appears to be a smirk of contempt and scorn, her assurances for elections sounds like guarantees for her perpetration, her boasting are a source of shame to Filipinos, and her arrogant claims to performance are insults to my feeble intelligence. Her frequent reference to the technicalities and rule of law, her conscious disregard of public opinion and dismissal of justifiable criticism – these gave me a weird, depressing feeling that we have made a grave mistake which we could not correct and we just have to live with it.

.

“I did not become president to be popular”, she sneered as if her record-setting, negative approval rating is a badge of honor – equating the people's gross rejection of her to a confirmation of her heroic and outstanding presidential performance. What a shameless presumption!

Excuse me Louisians, I think I need to go to the washroom and puke!

.

.

.

Childhood Memories in the Barrio

Kalburo Nights
by icarus

Tatang, my grandfather whom I grow up with is not only an outstanding farmer but is also a fine fisherman. When he goes to the farm to tend to his ricefields, he always had with him, strapped on his waist, his alat a woven bamboo basket with a narrow neck plugged with a cleverly devised cone of flattened bamboo sticks that allows things to slide in but not out. The things he put in his alat includes dalag, paltat, araro, dakumo, bisokol, leddeg, tukak not necessarily at the same time or in any particular order. He does not intently spend his time looking for these but in the bygone days, these species were plentiful in the Masicampo and the Pagumpias, two large tracts of open rice land in Asingan that borders Dupac, respectively on the east and on the west. These rice lands are also rich fishing grounds. A puddle in the paddies always had some stranded dalag or paltat. An old footprint in the rice field teems with leddeg and bisukol. An opening in the tambak would indicate a burrowed dakumo or two. Underneath heaps of mown rice stalks are frogs seeking shelter. My Tatang, wise on the ways of farm survival, has an uncanny instinct for this things and they invariably end up inside his alat.

.

Sometimes on a rainy afternoon, he would send me to town to buy kalburo (Calcium Carbide). Astride that big old fashion bicycle, I sit on the bike’s frame rather than the saddle which is too high for a young boy. I would pedal my way to Lua’s Variety Store some 4 km away, my butt obscenely swaying from side to side, my torso rhythmically forming and reversing in S-shape, as I strain to reach the pedals. Wet and muddy from racing with chasing dogs, I would insert myself inside the crowd of buyers lining the counters. Not wanting to get in contact with my armor of mud, they would unwittingly part to allow me to buy ahead - half a kilo of those gray lumps from the old Chinese grocer. On the way home, the same dogs would be waiting in ambush, raring to get their revenge from being outraced earlier - but that’s for another story.

.

Mixed with water, kalburo gives off acetylene, a combustible gas that fires my tatang’s olden bronze lampa giving off a bright yellow light that reflects brightly on its shiny parabolic reflector. Lampa is the local term for a carbide lamp consisting of two cylindrical bronze chambers threaded to each other; and a flame nozzle centered on a reflecting dish. The lower chamber holds the carburo; the upper chamber holds water and had knobs to regulate the flame. That’s what the old folks use before there were any flashlights.

.

When the rain intensifies in the evening, Tatang would load his lampa, put on his bistukol and annanga, strap on his alat, grab his tallakeb and walk right into the dark rainy night. He would be preceded by an ellipse of illuminated ground courtesy of the kalburo-fired lampa. All over the wide expanse of the masicampo and the pagumpias one could see solitary spots of lights moving about, piercing the black darkness of the pouring monsoon. They are from the lampas of hardy fish seekers and frog hunters called mannilaw just like my tatang. Fish and frogs and other fresh-water denizens are particularly friendly on rainy days, especially during heavy downpours. Their eyes shine like embers from the reflection of light. Caught in the focused brightness, they froze on their tracks like stones waiting to be picked up. Like clockwork, Tatang always comes back after an hour or so, his alat filled to the brim with tukak, pellat, dalag or whatever is in season. The catches are kept in separate burnays . What we cannot consume, my Inang would sell in the market the following market day.

.

But one night, my Tatang could not return within the expected time and my Inang was worried and concerned. His share of the evening snack – two ears of boiled corn – lay cold waiting for him. We can usually spot him approaching from 5o meters even in the heaviest monsoon. But an hour passed, two hours, three hours going four - still no sign of an approaching lampa. That night, after supper he listened to the chorus of frogs croaking all around and determined that they’re more plentiful in the Pagumpias, so off he went to that direction. My Inang is about to go out and seek for help when Brownie our old dog barked incessantly, facing the direction of the Pagumpias; yet I could see no lampa coming. He kept barking until his bark turned into an excited whine and then I was relieved to see my Tatang emerge from the hedges of saluyot just after the bamboo thicket. “Tatang, tatang…”, I ran out to met him in the drizzle. He fondly shielded my head with his wide palm as we walk to the house. Gone is his lampa and he has no alat. No tallakeb or bistokol, only the remnants of his tattered annanga. “What happened to you, you kept us all worried?” my Inang anxiously asked. “Oh, you wont believe me if I told you, Baket. Please hand me those dry clothes first. I’m freezing.” He dries up and changed clothes.

.

“So, what actually happened?”, Inang persevered. “And where is your catch?” “I don’t know, my alat might have dropped somewhere, Im not sure…I got lost in the dark…“ , stammered my brave and usually self-assured tatang. I can see from his face a trace of distress like he was stunned or had just come off a bad dream. And very tired.

.

“The catch is plentiful tonight”, he started. “I was just across the “kalungkong” about a hundred meters from here and already, my alat is half full. I was thinking to get back early when my lampa went off because of the strong wind. Not wanting to go home with half a basket, I tried to reignite the lampa with the few matchsticks that I brought. But each time the stick lighted it quickly gets snuffed out by the driving wind - until I ran out of matchsticks. I looked around. There were many lights from fellow mannilaw but they were too far off. But there, about a hundred meters to the southwest, I saw the light of another mannilaw. Quickly, I grabbed my tallakeb and proceeded towards his direction. But I could not just walk in the middle of the newly planted rice paddies, I have to follow the tambak. By the time I got to where he stood just a few minutes back, he has gone somewhere else. I was not closing the gap at all so I groped and walked even faster in the dark, chasing after the fellow mannilaw. Sometimes it gets bright and sometimes it disappears from view. But there it is in the general direction of southwest. I half run and half stumbled. The hundred meters become eighty and then sixty, then forty…! At last I could light up my lampa and continue with the catch… Thirty meters, twenty, fifteen…”Can I have a light, my friend?”, I called his attention in between my pantings. Only ten meters now. It was so dark but I was closing in very fast! And then the light stopped moving as if waiting to give me a light. Ah, the fellow heard me, at last. I hurriedly groped my way to the stationary light, and touch the flame with my lampa’s nozzle. It lighted with a zap. “Thank you my friend”, I mumbled. I was so grateful as my eyes adjust to the brightness only to realize that no one was holding the candle… Candle?!! I turned my lampa to my right and saw an arched window. I beamed my lampa around. More arched windows… and brick walls… and cornice and old stained glass panels. And then I realized! I was alone inside the Ermita, that old abandoned chapel in the middle of the “kamposanto” – in the graveyard!”

.

My Tatang tried to scream but no sound came out. He just run and run and run! He’s been had by the mangiyaw-awan. He never tried to recover his missing lampa. And he never bought a new one either. It was a good excuse not to go night-fishing again. Ever!

.

.

.

.

.

.

Is There Math in Drawings?

Not too long ago, I was into Software Development – specifically for software used in Structural Engineering Analysis and Design. Most of the algorithms and procedures involving engineering principles were more or less documented and explained in my published book, in my lecture notes and in the user's manual that comes with the software. However, there is a certain aspect on my computer programs that were never discussed nor formally documented because it does not relate directly with the field of application of the program.

I am talking about the Graphical Interface of the software. This is a module of the program which allows the user to say, describe the geometry and loadings of a building frame or a bridge truss by drawing it directly on the screen. An alternative option which we used in the more primitive versions is to type-in the coordinates of the joints and then describing the topology of member-joint interconnection by means of numerical inputs, etc. Such an option is tedious, error-prone and boring. Graphical Interface makes data entry a lot faster, more accurate, more fun and very intuitive. Computer Graphics also allow a more concise way of Input and Output presentation.

But Writing the visual interface is not easy. It is as challenging as writing the matrix algebra and finite-element portion of the program and therefore deserves as much attention and documentation.

This article, touches on the basics of Computer Generated Images which I used in writing the Graphical Interfaces on my programs. It is by no means exhaustive. You could e-mail me if you need to know more about this fascinating subject.

The Mathematics of Orthographic Projections and Linear Perspective

by icarus

Technical Drawing is one of the basic subjects in any engineering course. One could never overemphasize its importance. Ironically, it is often taken for granted - considered as a minor subject which is geared more to developing an engineer’s manual skill rather than his mental prowess. Thus, I will not be surprised if people, including engineers, would be intrigued by the article’s title and would ask: "Is there mathematics involved in Drawing?" I raised this same question once to my engineering students and the answer I got is: "yes, but only in scaling and dimensioning..." . Well, they’re partly right but they’re mostly wrong. Drawing is as mathematical as any civil engineering subject like say, geodetic surveying!

A drawing, the technical variety in particular, is nothing but a representation of a 3-dimensional object (with width, height and depth) and its location relative to the observer - into the two-dimensional realm of a flat surface, whether it’s the blackboard, a sketch pad or a computer monitor. Thus, going by this definition, drawing is essentially transforming the 3D coordinates of the corners of a real object into their 2D coordinates on the drawing surface, plotting these points and then connecting the dots with lines to represent the edges. To illustrate how point coordinate transformation is done from 3D to 2D, I could locate the hanging lamp in my room by its distance x from the side wall, distance y from the front wall and its distance z from the floor. However, in a perspective drawing of the room, the same bulb could be located in the sketch pad by defining the measurement u from the side and measurement v from the bottom of the sketch pad. Any other point inside the room could also be uniquely located in a similar fashion, transformed from its 3D location to its 2D position within the drawing frame.

Picture Plane and Orthographic Projections

The picture plane is a very useful concept in the process of making technical drawings. By definition, It is an imaginary transparent plane that the observer sets between him and his subject so that the outlines and details of the object could be traced by projection. It might as well be a window, or a rectangle formed by the outstretched thumbs and forefingers of both hands like when you’re preparing to take a photo shoot.

Figure 1 below shows a 3D solid, typical of building blocks or machine parts that we often encounter in engineering. If a picture plane is set directly in front of it, we could imagine parallel light rays reflected from the object to the picture plane allowing us to trace its “front view” as shown. Similarly, if the picture plane is transferred to the right of the object, a similar projection called the “right-side view” could be traced as in Figure 2. Now, notice that in both drawings, all corners of the object were projected to the picture plane but only a few of these show in the resulting drawings. That’s because some points are behind other points relative to the picture plane and thus, their projections would actually coincide with other points. This is what happens when you reduce the dimensions from three to two - the dimension normal to the picture plane is suppressed.

Figure 1. Front View Projection of Object on a Picture Plane

Figure 2. Right-Side View Projection of Object on a Picture Plane

We may generalize, intuitively, that to reduce a three-dimensional object into its planar projection, we simply have to suppress the dimension normal to the picture plane. Thus, a point which is located at P(x,y,z) on three-dimensional space will plot as P(u,v) in the picture plane by a very simple transformation: (u,v) = (T).(x,y,z); where T is a transformation matrix converting the 3D coordinates into planar 2D coordinates.

Explicitly for the “front view”, the y-coordinates are suppressed, thus any point located at (x, y, z) on space will now be plotted as a point (u,v) on the drawing pad. The conversion is done simply for each point as:

Similarly for the” right-side view”, the x-coordinates are also suppressed by using a different but similar transformation matrix:

We can derive a similar transformation matrix if you require a “top view” or any other view. Thus, we can go about placing the picture plane at any side and at any angle and inclination to obtain any side or angled view we want of the object so long as we can formulate the transformation matrix. Unfortunately, the transformation matrix is easily formulated only for cases where the picture plane is at right angles with the main axes of the object. For other angles and inclination, a rigorous expression is required.

But fortunately, there is an easier way. Instead of the observer going around the object, the observer and his picture plane could be fixed in one place, while we allow the object to rotate about any and all 3 axes! Thus, we can use the same 3D to 2D transformation matrix each time because we will be viewing it from a frontal picture plane all the time!

Coordinate Transformation through Rotations of Reference Axes

Suppose than from an initial upright and squared position we start rotating the object by angle A about the z axis, by an angle B about the x-axis and by an angle C about the y-axis; and then start measuring the new coordinates from the same set of axes as when the object is upright and squared. Then the object’s corners will have new coordinate triplets but essentially, the object has retained its shape. We can set-up the picture plane normal or perpendicular to the y-axis as before so that we can draw the projection simply by suppressing the y-coordinates using equation 1. Then we will be tracing its “front view” on the picture plane as usual - only that this time, the front view will be showing a different face.

Figure 3. Front View Projection with the object rotated about the 3 axes but the Picture Plane remaining in the Frontal Position which is parallel to the xz plane and normal to the Y-axis

Before we can do that however, we should be able to transform the coordinates of the object in its rotated state as in Figure 3. We start by considering a single point, P in space as shown in Figure 4. Its location in space is defined by the distances xo, yo and zo along the mutually perpendicular axes Xo, Yo and Zo respectively.

Figure 4. Transformation Of Coordinates by Successive Rotation Of Axes

Notice from the above figure that the same point P could be as accurately located if it were referred from say the X3-Y3-Z3 set of axes. The distances x3, y3 and z3 could be taken as the original coordinates of the point prior to rotating the object. Our goal now would be to find a way of converting the triplet (x3, y3, z3) into its equivalent triplet (x0, y0, z0) so that the coordinates will be normalized with respect to the picture plane. Once it is normalized, we can use equation 1 to produce a frontal projection simply by suppressing the yo values.

To arrive at the X3-Y3-Z3 position from the Xo-Yo-Zo position, we first rotate the Zo axis by angle A to produce the X1-Y1-Z1 position; then we rotate the X1 axis by angle B to make the X2-Y2-Z2 position; and finally, we rotate the Y2 axis by angle C.

Let us take the rotations one by one.

Taking the Xo-Yo plane after rotating the Zo axes by q as shown in Figure 5, we can relate the triplets (x1, y1, z1) with (x0, y0, z0) by simple trigonometry.

Figure 5. Relating (xo, yo, zo) with (x1, y1, z1).

From the above figure, we can deduce:

xo = x1· cosA - y1· sinA
yo = x1· sinA + y1· cosA
zo = z1
which when written in matrix form becomes:


Similarly, we take the Y1,Z1 plane after applying a rotation angle B about the X1 axis as shown in Figure 6, below.


Figure 6. Relating (x1, y1, z1) with (x2, y2, z2)

From which we can figure out the relationship:
x1 = x2
y1 = y2·cosB - z2·sinB
z1 = y2·sinB - z2·cosB

which in matrix form becomes:


Finally, we apply the rotation, angle C about Y2 axis to produce the X3-Y3-Z3 axes. Taking the x2-z2 plane as shown in Figure 7 below, we can derive the relationship as follows:
Figure 7. Relating (x2, y2, z2) with (x3, y3, z3)

x2 = x3·cosC - z3·sinC
y2 = y3
z2 = x3·sinC + z3·cosC

which in matrix form becomes:


If we substitute Eqn. 5 into Eqn. 4; and then the resulting expression is substituted into Eqn. 3, we finally get the relationship between the Rotated Axes, (X3, Y3, Z3) and the Normal Axes (Xo, Yo, Zo).


The 3D triplets (xo, yo, zo) are then transformed into the 2D pair (u, v) by Eqn. 1 which we repeat here:

Thus, we have generalized the projection of any object whose orientation is defined by any combination of values of angles A, B, and C.

Physical Interpretation Of The Angular Values A, B, and C.

The angle, A is the Rotation or Turning Angle by which we rotate the object while we keep it upright. Varying the value of this parameter has the effect of allowing the observer to go around the object.

The angle, B is the Tilt Angle or Elevation Angle which tips the object forward or backward. Varying this parameter has the effect of allowing the observer to go over or under the object.

The angle, C is the Slant or the Lean and measures the angle by which the object leans either to the left or to the right. Varying the value of this parameter is the equivalent of the observer's tilting his head on either side or for the photographer to shoot with a diagonally held camera.

Now, its time to try our little drawing algorithm. I have written a simple VB Program which "reads" the coordinates of the object in 3D and the connectivity of the lines which defines the edges. It then transforms these 3D coordinates into their 2D counterparts. Using the transformed coordinates, it plots the corners of the object on a window and then connect these points with straight lines, establishing the shape of the object in 2D. By varying the values of each angular parameter A, B and C, we will be able to draw different faces of the object.

For example, if we wanted to draw the Front View, we simply set the following parameters: A=0, B = 0, and C=0


To draw the Right-Side View, we set the angular parameters to:
A=-90, B = 0, and C=0



To draw the Top View, we set A = 0, B = 90, C = 0


If we set A = -30, B = 0, C = 0 then we obtain the Elevation View, from a horizontal vantage point 30 deg. to the right.


If we want a Worm’s View, from a vantage point 30 deg. to the right and 10 deg lower than the object, we should set A = -30, B = -10 and C = 0


Now, if we wanted to obtain the true isometric view, we have to turn the object to the left by 45 degress and make the object tilt forward by the arctan of (sin30/cos45) = 35.26 degrees and thus, the following parameters: A =-45, B = 35.26, C = 0


Why B = 35.26 degrees? I leave that as something for the reader to prove and ponder on.

Now, suppose that we are viewing from the same angles as in the isometric position, how will the object appear to us if we tilted our head by 15 degrees to the right. So then we set
A =-45, B = 35.26 and C = 15

As you can see, this mathematical method is more powerful and far more versatile than the graphical method of projections which uses pen, paper, T-squares, triangles and straight edges. It also uses exactly the same technique and procedure whether one wants to draw a front view, a side view, a bottom view or an isometric view because in fact they are all Front Views being Frontal Projections of the object on Normalized Coordinates.

But, it doesn’t stop there! When we made our Frontal Projections, we assume that the projection rays are parallel. This is very nearly true when, the viewer is infinitely distant from the observed object or when the latter is of relatively small dimensions. Actually, in most cases, the projection rays are far from parallel - especially if the observer is relatively near the object.

Perspective Projections

Orthographic Projections are fine for Technical Drawings where it is important to have consistent dimensional scale of measurements. However, Orthographic Projections present images that are not as "realistic" as they actually appear to us. A more realistic image or drawing could be attained if we take into account the fact that the projection rays converges to the observer rather than stay parallel. To illustrate, suppose that we set the Picture Plane at a distance "a" from the object and that we stand along the same line at a farther distance, D. This set up is illustrated in plan and in elevation by the figures below:

From the above figure, we chose a typical corner of the object which is labelled as P. Now observe how the image of P projects to the camera or eye of the observer along the blue dotted straight line which pierces the Picture Plane at point P'. In fact, all other points on the object project in an identical manner. Thus, the location of the plotted points on the picture plane is not only a function of the xo and zo coordinates (how far to the side and how high) but also yo (how far back). Using similar triangles:

u = (D)(x)/(D + a + y)
v = (D)(z)/(D + a + y)

we can rewrite this more concisely in transformation matrix format as:


which is identical in form to the Frontal Projection transformation (eqn. 1) but instead of 1's we used the argument, T which is equal to

T = D / (D + a + y0)

The View Angle and the View Distance

The human eye covers only a certain region at any one time. It can see only objects or parts of objects that fits within the "cone of vision". The angle by which the cone of vision opens is called the View Angle, W. The wider the angle, the larger the coverage. To be able to see the entirety of an object, the observer has to be at a certain distance from the object so that the observed object fits within the cone. The distance is called the View Distance, D. The wider the View Angle, the less is the View Distance required. The relationship between View Distance and View Angle is defined by the equation

D = Smax / (2tan(W/2)) - a

where Smax is the maximum cross-sectional dimension. Since, the sizes of objects being viewed varies greatly, the View Distance also varies accordingly. Therefore, it is usually more practical and convenient to define the View Angle instead. Moreover, the view angle has more or less established values. For the normal human eye, W ranges from 30 to 60 degrees. For the lens of a standard camera the View Angle is about 50 degrees. Wide Angle cameras have lens angles of W = 90 to 180 degrees (fish eye lense). Macro as well as telephoto cameras have view angles usually less than 25 degrees. The distance "a" could be taken as zero, meaning that the picture plane is taken is placed within the object. More reasonably, we can place the PP just in front of the object by letting

a =1/2(xmax^2 + ymax^2 + zmax^2)^0.5

where xmax, ymax and zmax are the maximum dimensions in the x, y and z directions.

By changing the Transformation matrix of eqn.-1 to the Transfomation matrix of eqn.-7, we get Perspective Views rather than Orthographic Views.

I would also like to call your attention over the fact that the concept of a "Vanishing Point", a fundamental concept in manually drawn perspective, has become totally irrelevant. Also, the mathematical method which was developed here do not make any distinction on one-point, two-point, three-point or multi-point perspectives - they are one and the same, depending on the orientation and view angle.

Thus, viewing the same object using a 50 degree view angle,W; A = -45, B = 35.26 and C = 0, we get this bird's eye view perspective.


Using a wide-angle lens with W=150 deg and rotational parameters of A=-15, B = -15, C = 0, we get a dramatic worm's eye view of a medium-rise building.


A normal view from the ground could usually be obtained by setting A = -50, B=-10, C=0 and W = 50 degrees as in the drawing below.


I used approximately the same angular settings A = -50, B=-10, C=0 with a 50mm lens camera to take this photograph of the majestic Our Lady Of Namacpacan Church in Luna, La Union.

So the next time somebody tells you that a graphic artist or a photographer knows nothing about Math, think again. He just might turn out to be a Math wizard.

Graduations and Other Highlights of my Vacation

Twenty two years ago, Marvin James, my second child and eldest son was born in the midst of our financial instability and uncertainty. I was freshly kicked out of my job from (Imelda Marcos') Ministry of Human Settlements as a result of the upheavals wrought by EDSA People Power I. Suzy and myself are both unemployed as she had to stay home full time to take care of our two kids. Being young and barely experienced engineer, I had a hard time competing for whatever engineering jobs there were during those crisis years. With no one willing to give me a break, I had to go back to where I had a chance at excelling - the academe. It was on my first year as an Engineering Instructor at Saint Louis that James first saw light. Despite the difficult times, I remember being very excited and extremely proud of my first son. In a way, he gave me hope and a new resolve to work harder to provide for their future.

Now, two decades later, my excitement and faith in my son is proven right. At last, I am taking my just reward. James have grown and matured into a fine young man - a Civil Engineer, albeit a neophyte. I guess any parent would be proud to have him for a son. Thanks mainly to my wife, Suzy who did more than a splendid job of raising my children almost singlehandedly while I worked away in Makati and now, overseas.

If there was nothing else but James graduation, it would still be more than worthwhile to be home this year. Here are some of the highlights of my vacation in pictures...

Just before the ceremonial march:

The two people who worked hardest for the realization of this dream,
M. James and my wife, Suzy.


Me, sharing in the limelight and luster of the moment...

Kami naman!!

Ivy and Monette got to have their pictures taken, too...

Isa pa, isa pa...! Just in case you missed how proud I was...

Ako din, ako din!! Ivy is agaw-eksena as usual...

With Lilibeth, his very special friend...

she's an industrial engineer.

With Beth's parents... Uhurm, looks like a preview of some event to come...

Picture...picture! After a simple (and a must) al-fresco dinner at Dencio's Grille

on the top floor of SM Baguio, overlooking the city of pines...

The morning after is not complete without a mass at the Baguio Cathedral...

Two of my babies, Rachelle and Ivy. Can you spot the other one

due in 5 months? hehehe...

Say cheese!!!

The two nurses in the family plus the perennial clown...

Dejavu:

Suzy and I posed the same pose on this same spot

some twenty three years ago.

Back then, she's all dressed in white...

Breakfast at good ole Rose Bowl...

The First Lady.

She looks every bit a lovely First Lady to me...
But she does not like to be called that...

'First' alludes that there is a second,

or a third... hehehe.

My Fair Lady would be fine, I guess.

with the President:

(of North Central School Grade V class of '71 hehehe).

Ninong at Ninang:

Here, were all dressed up for a very important event...
We stood as principal sponsors for the umptenth time. This time,

to the newly-wed couple, Engr. and Mrs. Christopher Angala.

Congratulations and best wishes to Chris and Marcie!

Two Ninangs: Also with us is Dean Carisa Blancas of the University of Baguio,

also a perennial favorite choice for a Ninang.

Remembering Jay-ar: We always make it a point to be with our dear departed every so often. Light moments with Monnete and Ivy as Marvin and Suzy looks on. Rachelle's taking

the picture behind the camera. The Forrest Lake memorial park in our town, San Juan,

is probably the best in LaUnion.

I cant remember what Ivy is so excited about. Here, she outdoes julia roberts for the world's widest grin!

Now I know what it was. Its the calesa ride in Vigan, that's what!

Its a common ride during my elementary days... Now, it has

become some kind of an oddity.

Melee at dinner time. All the best places in the world,

could never be near as fun as being home sweet home...

Already looking forward to my next one.

A Sequel To My Everlasting Chrysanthemum

Bring Me Back That Summer
by sammy antonio

Who wouldn't miss the sunny day full of life during summer,
When birds are free to fly, even the night owl slept in slumber;
The night sky is glittered with stars in perfect number
Leaving us to dream, tightly holding our hands together.

We used to fly kites, on that verdant shore of greens;
Holding the string in unison, our kites soar high toward our dreams.
Then a strong wind blow and broke that man made string,
The kite flew away, so with you, leaving me sad, devastated and in vain.

Another summer season had passed, my heart still in tears;
Longing for those days when your vibrant laughter filled the air.
I could hear it, 'til your silhouette gradually appears;
I know you're breath away, but in my thoughts you're near.

And now the dark cloud of uncertainty looms along the horizon ,
Don't know where this lonesome journey leads me, as i travel alone.
.
.
.
.
.

Solution To The Puzzle For The Moderator

As the title of the puzzle suggests, the challenge was directed on me. But I never had the chance to try if I could handle it or not. Most probably not. Fortunately, two former students of mine stood up and save me from this puzzle quandary. Engr. Jeffrey Agbunag and Engr. Ramil Balisnomo belong to the last batch (and therefore youngest) of the CE students I handled for Reinforced Concrete Design. They now work in Manila as Structural Design Engineers and are planning to join us in Dubai in the near future.

For further clarity, I made minor alterations to their solution and added some explanations without changing the essence of their reasoning. So here it is.

A Puzzle For The Moderator

There are three positive integers, each with two non-repeating-digits. The sum of the first and the second integers when added to thrice the third integer results to 242. When half of the second integer is subtracted from a number which is the reverse of the first integer, and the difference is added to a number which is the reverse of the third integer, we obtain 116. If we reverse the second integer and deduct the resulting number from the first integer, we get a difference of 10.

Amusingly, the third integer has other amazing properties. When the number and its reverse is divided by two, the halves are reverses of each other. The halves when further halved are also reverses. Can you give me the integers?

The Solution:

by Jeffrey Agbunag and Ramil Balisnomo
.
Let :
A = first digit of the first integer
B = second digit of the first integer
C = first digit of the second integer
D = second digit of the second integer
E = first digit of the third integer
F = second digit of the third integer
Then:
10A + B = first integer; 10B + A = the reverse of the first integer
10C + D = second integer; 10D + C = the reverse of the second integer
10E + F = third integer; 10F + E = the reverse of the third integer

From Condition 1:
(10A+B)+(10C +D) + 3(10E+F) = 242

From Condition 2:
(10B + A) – 0.50(10C + D) + 10F + E = 116

From Condition 3:
(10A + B) – (10D + C) = 10

The three equations when simplified and written in matrix form becomes:

The non-square coefficient matrix will show that no unique solution is possible because there are only three equations and there are six unknowns. We would therefore require three more independent equations. In order to produce the extra conditions, the duo came up with two remarkably original propositions:

The second paragraph of the puzzle talks of the third integer and its reverse having their halves and quarters as reverses. Thus,

1. The Agbunag Theorem:

“For the set of positive, single digit numbers, two - and only two - could be halved twice and still produce whole number results.” These are the numbers 4 and 8. Thus, they concluded that only the two-digit numbers 48 and 84 could satisfy the conditions of the second paragraph. However, they correctly pointed out that it could not be 84 because it would contravene the stipulations of Condition 1 (the sum would exceed 242 since 3 x 84 is already 252). Hence, the third digit is 48 and therefore

E = 4 and F = 8.

Adding these two equations to the matrix expression, we get

Still, there are more unknowns (6) than equations (5). So from where could we possibly acquire the 6th equation?

Enter Jeffrey’s and Ramil’s numerical acuity; and they come up with a simple but universal truth where most of us could only see a blank wall.

2. Balisnomo’s Theorem:

“If the difference between two 2-digit numbers is 10, their last digits must be the same.”

Thus, hidden in Condition 3, they pointed out that B must be equal to C by Balisnomo’s Theorem. In equation form:

B – C = 0

which when inserted into the matrix expression, we finally obtain an invertible coefficient matrix.

Explicitly, we obtain and write the inverse and perform the matrix multiplication, using Excel:

to obtain:

finally,

1st integer = 36
2nd integer = 62
3rd integer = 48

sana tama sagot namin. Hehehe.

God bless.

-Jeffrey and Ramil
.
I have confirmed this with Carlo. TAMA ang sagot niyo! Congratulations and my admiration to both of you.

- icarus.
.
Solutions for the other Problems:

A certain Marlon solve two out of the three problems which I posted as additions to Carlos lone puzzle. I am not sure if Marlon is a Louisian but he is one heck of a logical guy. I hope he will come forward and identify himself because he is surely a welcome addition to our rank. Here are the postings he made on the comment sections:

Marlon said...

Given : ABCDEx 4 = EDCBA
The answer is : 21978x 4 = 87912

But rather than just give you the answer, here's how I figured it out.

First, it is obvious that A must be an even number, because we are multiplying by 4 (an even number). The last digit will therefore be even. It can't be 0, because that would make ABCDE a four-digit number. It can't be more than 2, because that would result in a six-digit answer.

So A = 2.

2BCDEx 4= EDCB2

So what can E be? The choices are E = {3, 8} because 3 x 4 = 12 and 8 x 4 = 32. But a value of 3 doesn't work in the result (3????) because it is too small.

2BCD8 x 4= 8DCB2

Since the final number is 8 and we have 2 x 4, that means there is no carry from the prior multiplication (4 x B + carry). So B can't be anything higher than 1, possibly 0.Looking at the other side of the equation, we have 4D + 3 = (a number ending in 0 or 1). In other words, 4D must end in 7 or 8.

Obviously only 8 works, because 4 is an even number. Working forward again, that means B = 1

21CD8 x 4 = 8DC12

So what values of 4D result in a number ending 8? 4 x 2 = 8, 4 x 7 = 28. Now 2 is already taken and the problem said the digits were unique. So D = 7.

21C78 x 4= 87C12

Finally, we have a carry of 3 (from 28 + 3 = 31). And when we calculate 4C + 3 it must also result in a carry of 3 and a last digit of C.

In other words: 4C + 3 = 30 + C

This is easy to solve:

3C = 27 ; C = 9

Thus the final answer is: 21978 x 4 = 87912

February 19, 2009 9:10 PM
Marlon said...

The secret is to notice that the answer has more letters (5 letters) than the question (4 letters).

That M at the beginning of money is a carry from the thousands place, so M = 1.

Now we have:

SEND+ 1ORE = 1ONEY

Now, in the thousands place there is a 1, so the only value for S that could cause a carry is S = 9 and that means O = 10.

Now we have: 9END+ 10RE = 10NEY

Now look at the hundreds place. If there were no carry from the tens place, E and N would be the same because E + 0 = N, but E and N can't be the same, so there must be a carry from the tens place.

Now we have: 1 1 <-- carry9END+ 10RE = 10NEY

Now the equation for the hundreds place is 1+E+0 = N or just 1+E = N. In the tens place we can have N+R = E+10 if there is no carry from the ones place, or we can have 1+N+R = E+10 if there is.

First test: no carry from the ones place:N+R = E+10 and 1+E = N(1+E)+R = E+101+R = 10R = 10-1R = 9

But S = 9, so R cannot = 9.

That means there is a carry from the ones place and we get:

1+N+R = E+10 and 1+E = N1+(1+E)+R = E+102+R = 10R = 10-2R = 8

So now we have: 1 11 <-- carry9END+ 108E = 10NEY

N cannot be 0 or 1 because 0 and 1 are taken. N cannot be 2 because 1+2+8 = 11 and then E would equal 1, but it cannot equal 1 because 1 is taken.

N could equal 3,4,5,6 or 7 but it cannot equal 8 or 9 because 8 and 9 are taken (and E must be 2,3,4,5 or 6 because it is 1 smaller than N).

If E were 2, then for the ones place to carry D would have to be 8 or 9, and both 8 and 9 are taken, so E cannot be 2 (and N cannot be 3).

If E were 3, then for the ones place to carry D would have to be 7,8, or 9, but D cannot be 7 because then Y would be 0, which is taken, so E cannot be 3 (and N cannot be 4).

If E were 4, then for the ones place to carry D would have to be 6,7,8, or 9. D cannot be 6 because Y would be 0, but D cannot be 7 because Y would be 1, and 0 and 1 are both taken, so E cannot be 4 (and N cannot be 5).

The only two possibilities for E now are 5 and 6. If E were 6, then N would be 7 and D would have to be 4 (which would make Y = 0), 5 (which would make Y = 1), 6 (which is taken by E), or 7 (which is taken by N).

There are no solutions for E = 6, so E must be 5.

So now we have: 1 11 <-- carry956D+ 1085 = 1065Y

Doing the same reasoning for D and Y and get the answer

9567+ 1085 = 10652

February 19, 2009 9:21 PM
ICARUS said...

Crisp and precise logic, nothing wasted, each line, each thrust and stroke goes straight for the jugular. Well done, Marlon!

I can tell you're a puzzle enthusiast yourself. Maybe you can send us some more interesting posers.

Thanks a lot.
.
.
.