Kwentong Hapon

For those who were born in the 50's and the 60's, war time stories were a staple. Neighborhood evening conversations almost always wind up to war-story telling. The movies were also full of them, almost promoting a blind hatred to anything Japanese; and worship of anything American. Ford was the revered auto brand, Toyota was a ridiculed counterfeit. Zenith is the loved TV name and the cheap inferior alternatives are Hitachi, Toshiba, National. Everyone is very conscious of the Made-in-USA vs Made-in-Japan thing. Now, it is altogether different as Japanese technology has caught up and have even surpassed their American counterpart. At the same time, the wartime stories are also pushed to the far side of our consciousness. The story below gives a glimpse of postwar barrio life in Dupac, Asingan, Pangasinan.

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Kwentong Hapon

by icarus

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Having spent my kiddy years in Dupac, I could recall vividly the moonlit summer nights when the folks in my neighborhood gather in small groups and sit around along the then dirt road to San Manuel. Sometimes, they bring out a “bangko” so they can sit comfortably in a row but most of the time they just squat on their haunches on the ground or sit on some big stone, some piece of wood or tree trunk; or on their rubber slippers on the ground.
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Early evenings, generally after supper, before going to sleep are spent this way. There were no TV’s then so these “ummok” or impromptu assembly is a way of catching up with the latest news, gossip or what have you. The day is spent away working in the farms or in some other places and thus, the early evening is a perfect time to relax and socialize with the neighbors. And while us kids are busy at patintero, tumbang preso, or taguan pong, the elders are on with their banters and idle conversations. Invariably, these talks - like we do now – lapse into nostalgia.
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Late in the 60’s and into the early 70’s when WWII was just a mere 25 years behind, people then don’t get tired of talking and reminiscing about the “Pistayem” (Peacetime), the “Bakwit” period (Evacuation), and the Liberation. Invariably, each has a favorite anecdote or misadventure to tell - some poignant, some heroic, and some are really, really funny. Of course, these were oft-repeated tales and we could almost memorize each plot and sequence but we never tire of listening and laughing at the often funny ending.
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Ever the clown, Laki Inong, otherwise known as “Patsara” (grasshopper) because of his tall and lanky frame, is the best story-teller of all – embellishing his tales with exaggerated bodily gestures and pantomime. When he starts telling his “Kwentong Hapon”, we abandon our tumbang preso and gather around him.

None of his many other war tales could bring the house down as much as this one. It goes like this:

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It was January, 1942 and Dupac was a ghost town. The residents were hiding in the Bakwit which is what they call the wartime hiding places on the then thickly-forested bamboo grooves a few kilometers into the northeast portion of Dupac. But to secure the properties and houses that were left behind in the barrio proper, a daily 3-man patrol was organized among the men in rotating shifts. On one particular duty shift, Laki Inong, Laki Pelis (My Grandfather) and Laki Mesiong, the team leader, were on patrol. They are proud to be among the few men brave enough to stay behind. Laki Mesiong was a landed school teacher and had some influence. He brandished with him a carbine semi-automatic. The Japanese have not arrived in town yet and people don’t really know what to expect of them. The trio is therefore in high spirits and they wore plenty of swagger, playing the de-facto sherrif's role to the hilt on the abandoned barrio.

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As they were whiling away their shift hours at the “bantayan” which is a makeshift shed of bamboo and cogon and two benches facing each other, they were boasting among themselves that if ever the japs get into town, they will each capture a soldier to be kept as a farm hand or servant. "Ammo da met la ngata iti agarado, pare?" wondered Lakay Pelis. "Isudan a iti pag-guyod ko iti arado no di da ammo iti aggiggem ti witiwit!" , Lakay Inong boasted.

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No sooner than their laughter died down when they heard the distinct drone of a big engined vehicle accompanied by the sound of metal grinding on stone. It was a military tank crawling slowly from the north... But Laki Mesiong quickly dismissed it as a friendly American tank being maneuvered for defensive position from the Binalonan Camp. Whatever he says, the other two believed and trusted completely, after all Laki Mesiong is friends with the Americans who gave him the Carbine. It was that reassuring to his men when he talked.

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Laki Inong and Laki Pelis were jumping and waving their hats in excitement as the tank approached from the north. But as it got nearer, the emblem of the Rising Sun becomes apparent. “Hapon sa met kaka?”, the two nervously uttered, praying not to get a confirmation from their team leader. “Hapon a nga talaga, Penong…!” came the trembling reply, unconsiously combining the names of his subordinates in his excitement.

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Suddenly, the tanks headlight were beamed on them, and in his shock, Laki Mesiong accidentally pulled the trigger, just missing Laki Pelis’ foot by the breadth of a hair. That started it. The Japs opened fire. “Taray Buatit tayon!” they exclaimed in panic as they were chased by machine gun fire from the tank. As 50-caliber bullets ricocheted in their wake, the three would-be-captors-of-Japs scampered like headless chickens through the thick thickets of prickly shrubs and thorny bamboos and never felt a thing!

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A Japanese soldier for a man servant, eh? That’s the farthest from their mind after that... Hahaha.

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.

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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

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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
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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.
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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.
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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.

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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.

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Mabuhay ang Pilipinas!
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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.

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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.
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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.

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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)
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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.

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Thank you and best regards to everyone.
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Very truly yours,

Engr. Isagani S. Espejo

Louisian 260162

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double-click image to enlarge:

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IMHO

SONA babitz!

by icarus

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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.

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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!

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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.

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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.

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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.

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“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!

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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.

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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.

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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.

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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.

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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.

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“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.

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“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!”

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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!

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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.