A general sectional volume equation for classical geometries of tree stem

Transcription

1 Madera y Boque 6 (2), 2: NOTA TÉCNICA A geeral ectoal volume equato for clacal geometre of tree tem Ua ecuacó geeral para el volume de la eccó de la geometría cláca del troco de lo árbole Gldardo Cruz de Leó ABTRACT Th work refer to the clacal theory of tree tem form. It how the dervato of a geeral ectoal volume equato for frutum of old of revoluto geerated by the fucto y 2 p x where, p a potve cotat, ad ay potve teger. The cylder cae preet a gular tuato becaue of t ectoal volume equato caot be defed for a t kow for the geeratg fucto. However, that geometry mplct a a trval oluto of the derved equato. The kow ectoal volume equato for frutum of parabolod, cood ad elod are partcular cae of that equato for, 2, ad 3, repectvely. The geeral ectoal volume equato ha a uexpected tattcal ature. It gve a a arthmetc mea of geometrc mea The clacal theory of tree tem form cotue beg preet the foret meauremet teachg ad reearch. Th work could cotrbute to mprove the udertadg o that theory. PALABRA CLAE: Dedrometry, appled mathematc. REUMEN Ete trabajo e refere a la teoría cláca de la forma del troco del árbol. e muetra la dervacó de ua ecuacó de volume geeral de la eccó de óldo de revolucó trucado geerada por la fucó y 2 p x dode, p e ua cotate potva, y u etero potvo. El cao del cldro cottuye u cao gular pue u ecuacó de volume de la eccó o e puede defr para, ya que e coocdo por la fucó geeradora. embargo, ea geometría etá mplícta como ua olucó trval de la ecuacó dervada. La ecuacoe coocda de volume de eccoe trucada de parabolode, coode y elode o cao partculare de la ecuacó para, 2 y 3, repectvamete. La ecuacó geeral de volume de la eccó e de ua aturaleza etadítca eperada. e da como ua meda artmétca de meda geométrca. La teoría cláca de la forma del árbol gue etado preete e la eeñaza de medcó e vetgacó foretal. Ete trabajo podría cotrbur a mejorar la compreó de ea teoría. PALABRA CLAE: Dedrometría, matemátca aplcada. Facultad de Igeería e Tecología de la Madera. Uverdad Mchoacaa de a Ncolá de Hdalgo. Apartado Potal 58, 58. Morela Mchoacá, Méxco. Phoe: (443) E-mal: [email protected]

2 9 A ectoal volume equato for geometre of the tree tem clacal mode INTRODUCTION Tree tem ad log volume etmato are ome of the ma goal foret meauremet. It a dffcult tak becaue thoe volume deped o ther geometry (Brack, 999). For more tha a cetury reearcher have bee workg extevely o the problem of tree tem geometry from fudametal ad emprcal pot of vew. I the frt cae oe le of aemet ha bee by mea of mechacal theore. A quoted by Laro (963) ad Dea ad Log (986), aumg that tree tem develop a partcular form to mata the bedg tre cotat uder the fluece of wd, 893, Metzger foud a tem geometry where the cube dameter proportoal to heght. Regardg that gravty determe tree tem hape the quare dameter reult proportoal to the cube of heght (McMaho, 973). The emprcal pot of vew ca be dvded o clacal ad curret theore. I the clacal theory a tree tem modeled by part ug mple geometre of old of revoluto (Grave, 96). Curret theore ugget partcular geometrc model developed for each pece by mea of taper equato for the whole tree tem (Wet, 24). The clacal theory of tree tem form ha a gfcat poto the foret meauremet lterature. At leat for a cetury t ha bee cluded tadard book of that feld (Grave, 96; a Laar ad Akça, 27). The clacal theory codered a part of old reearch foret meauremet (Wet, 24). However, t cotue beg a ueful referece reearch problem that deped o tree tem geometry. For tace, t ha bee ued to derve the equato for the cetrod poto the developmet of the cetrod amplg method for tree tem volume etmato (Wood et al., 99) ad to derve a geeral relato betwee tem volume ad tem urface form-factor depedet of poto (Ioue, 26). Th work how the dervato of a geeral ectoal volume equato for the clacal tree tem geometre. It baed o the clacal theory of tem form, ectoal method of volume etmato, elemetary algebra, ad calculu. A ummary of that methodology provded below. MODEL AND METHOD Clacal geometre of tree tem Tree tem form ca be modeled by logtudal ecto ug elemetary geometre of old of revoluto geerated by the equato y 2 p x [] where p ad are cotat uch that p > ad (Grave, 96). Curretly, oly the cae,, 2, ad 3, related to cylder, cood, parabolod, ad elod, repectvely, are aged the clacal tree tem geometre (Déguez- Arada et al., 23). For th work, the orgal cocepto of equato [] wll repreet the clacal geometre of tree tem ad the tudy refer to ay potve teger. Tree tem volume etmato Dfferet type of method to etmate tree tem volume are ued foret meauremet. ome of thoe method cot : ) to propoe a taper equato ad the to etmate the volume by tegral calculu, ) to propoe a volume equato where the volume ca be drectly obtaed a a fucto of heght, ormal dameter, ad total heght, ad )

3 Madera y Boque 6 (2), 2: to ue a ectoal method where the volume ca be kow for logtudal ecto a fucto of cro ectoal area ad legth (Brack, 999). The problem of volume etmato crtcal whe the volume of log eed to be etmated at feld where o formato of the orgal tree tem tadard parameter or geometry at had. For thoe commo tuato the ue of ectoal method uavodable (Bruce, 982). Geeral defto of ectoal method For mplcty, tree tem logtudal ecto ad log wll be called here tree tem egmet. Ay ectoal method to etmate the volume of a tree tem egmet, of legth L, refereed to the volume of a cylder ad ca be wrtte a L [2] where a average cro ectoal area (Avery ad Burkhart, 22). What defe a partcular ectoal method the form of a fucto of elected cro ectoal area. Equato [2] wll be eetal th work. tadard ectoal method The geerally accepted ectoal method for volume etmato of tree tem egmet the foret meauremet lterature are the Huber, mala, ad Newto method. Followg the otato of Chapma ad Meyer (949), for a tree tem egmet, of legth L, ed cro ectoal area, B at the large ed, b at the mall ed, ad B /2 at the mddle, thoe tadard ectoal method ca be repectvely defed by N H B / 2 B + 2 b B + 4B / 2 + b 6 [3] [4] [5] The, H H L, L, ad N N L, are ther correpodg volume agreemet to equato [2]. Defto of ytem ad parameter for th work Uually, old of revoluto geerated by equato [] are placed a x y z Cartea coordate ytem wth ther ax of rotato alog the x ax ad ther tp at the org (Déguez-Arada et al., 23). The ytem for th work are defed a frutum of legth L of thoe old of revoluto for ay potve teger. The ed cro ectoal area wll be called here, for the mall ed, at x x, ad for the large ed, at x x 2. By defto, x <x 2, the, L x 2 -x. ectoal volume equato for the tree tem clacal form The volume for frutum of old of revoluto geerated by equato [], for,, 2, ad 3, a fucto of ed cro ectoal area ad legth, are L L + L 2 [6] [7]

4 92 A ectoal volume equato for geometre of the tree tem clacal mode L L 4 [8] [9] for cylder, parabolod, cood, ad elod, repectvely (Grave, 96). I agreemet to equato [2] ca be defed for equato [6]-[9]. For cylder,. For frutum of parabolod, cood ad elod, ther mea cro ectoal area are repectvely [] [] [2] 4 I the followg, t wll be how that equato [7]-[9] are partcular cae of a geeral ectoal volume equato related to the geeratg equato [] for ay potve teger. The cae deerve a partcular dcuo gve at the ed. REULT Dervato of a geeral ectoal volume equato for clacal geometre of tree tem The ma reult of th work how the form of a mathematcal theorem. Theorem If the volume for a old of revoluto frutum, of legth L, ad ed cro ectoal area at x x, ad at x x 2, geerated by the equato y 2 p x, where a potve teger, the, ca be expreed the form + [3] Proof If the geeratg fucto for a old of revoluto gve by y 2 p x the the volume for t frutum from x x to x x 2, gve by tegral calculu, π p (tewart, 22). The relato [4] [5] well kow the mathematcal lterature (pegel ad Moyer, 27). Gve that L x 2 - x ad, from equato [], equato [5] take the form x + 2 x + L ( x2 x ) ( x2 x) x2 x / ( ) / ( ) x2 / π p x /π p + + L ( ) / / ( x2 x ) πp [6]

5 Madera y Boque 6 (2), 2: ubttutg equato [6] equato [4], reult ( ) / + whch exactly equato [3]. DICUION L [7] The geeral ectoal volume equato [3] ha bee derved for old of revoluto geerated by the equato y 2 p x for ay potve teger. It ca be ealy how that, for, 2, ad 3, the ectoal volume equato [7]-[9] for frutum of parabolod, cood ad elod, are repectvely recovered. Alo, for, baal area, ad LH, ther total volume, H/(+), are obtaed. However, a t ca be ee equato [3] a ectoal volume equato for the cylder cae caot be defed. Neverthele, t ot eceary becaue the geeral equato reduce to the cylder volume equato whe for ay. That codto of cylder volume recovery eem to be eceary for ay ectoal volume equato. I a propoal to evaluate varou ectoal method to etmate butt log volume Bruce (982) take that codto a a gude to elect them. Let u udertad the meag of equato [3]. I agreemet to equato [2] t ca be wrtte the form L where hould correpod to a geeral mea cro ectoal area. I partcular, for / /( + ) equato [],, the arthmetc average of the ed cro ectoal area ad. For equato [], 2 the arthmetc average of ed cro ectoal area,,, ad ther geometrc mea (Uraga-aleca, 28). I geeral, f X, X 2,... X, are the poble value take by a varable X, for a ample of ze, ther geometrc mea a meaure of cetral tedecy defed by X X 2 X (Chapma ad Meyer, 949; a Laar ad Akça, 27). The term for,,2,...,, repreet the geometrc mea cro ectoal area for, X X 2... X- ad X -+ X -+2. X -+2 X -+ X. The, the geeral mea cro ectoal area /( + ) correpod to the arthmetc average of thoe geometrc mea. Although the problem for th work wa ot orgally defed a of tattcal ature t tured o a tattcal oe whoe complete aaly progre. A addtoal property of equato [3] that t repreet a ymmetrcal fucto o ad, for ay, becaue of what mea that egmet volume depedet of t oretato alog t ax of rotato. Alo, the equato for the tadard ectoal method obey that ymmetry property. That codto could be aother feature to take accout the propoal of ew ectoal method for volume etmato. The mot mportat cotrbuto of th work the geeral ectoal volume equato [3] from what kow reult of the foret meauremet feld, ce more

6 94 A ectoal volume equato for geometre of the tree tem clacal mode tha a cetury (Grave, 96), are partcular cae. ACKNOWLEDGEMENT The author deeply ackowledge Profeor Adra Goodw for h helpful uggeto. Th work wa upported by Coordacó de la Ivetgacó Cetífca de la Uverdad Mchoacaa de a Ncolá de Hdalgo. Morela Mchoacá, Méxco. REFERENCE Avery, T.E. ad Burkhart, H.E. 22. Foret Meauremet. Mc Graw Hll. UA. Brack, C Foret Meauremet ad Modelg [ole]. Avalable from rea o c a t e d. a u. e d u. a u / meurato/volume.htm. Bruce, D Butt Log olume Etmator. Foret c. 28(3): Chapma, H.H. ad Meyer, W.H Foret Meurato. McGraw-Hll. Edto. UA. Dea, T.J. ad Log, J.N aldty of Cotat-tre ad Elatc-tablty Prcple of tem Formato Pu cotorta ad Trfolum pratee. Aal of Botay. 58: Déguez-Arada, U., Barro-Ata, M., Catedo-Dorado, F., Ruíz-Gozález, A.D., Álvarez-Taboada, M.F., Álvarez- Gozález, J.G. y Rojo-Albareca, A. 23. Dedrometría. Mud-Prea. Madrd, Epaña. Grave, H Foret Meurato. Joh Wley & o. UA. Ioue, A. 26. A model for the relatohp betwee form-factor for tem volume ad thoe for tem urface area coferou pece. J. For. Re. : Laro, P.R tem Form Developmet of Foret Tree. For. c. Mo. 5. McMaho, T ze ad hape Bology. cece. 79: pegel, M.R. ad Moyer, R.E. 27. Algebra uperor. McGraw-Hll. Méxco. tewart, J. 22. Cálculo Tracedete Tempraa. Thompo Learg. Méxco. Uraga-aleca, L.P. 28. Aplcacó del Modelo Cóco egmetado a lo Tpo Dedrométrco Cláco como Fucó del Número de egmeto. M.c. the, Facultad de Igeería e Tecología de la Madera, Uverdad Mchoacáa de a Ncolá de Hdalgo, Morela Mchoacá, Méxco. a Laar, A. ad Akça, A. 27. Foret Meurato. prger. Netherlad. Wet, P.W. 24. Tree ad Foret Meauremet. prger-erlag. Berlí. Wood, G.B., Wat, H.. Jr., Loy, R.J. ad Mle, J.A. 99. Cetrod amplg: A varat of mportace amplg for etmatg the volume of ample tree of radate pe. Foret Ecology ad Maagemet. 36: Maucrto recbdo el 3 de julo de 29 Aceptado el 8 de eero de 2 Ete documeto e debe ctar como: Cruz de Leó, G. 2. A geeral ectoal volume equato for clacal geometre of tree tem. Madera y Boque 6(2):89-94.