By Coble A. B.
Read or Download Geometric Aspects of the Abelian Modular Functions of Genus Four (I) PDF
Best geometry and topology books
From Geometry to Quantum Mechanics: In Honor of Hideki Omori
This quantity consists of invited expository articles by way of recognized mathematicians in differential geometry and mathematical physics which were prepared in social gathering of Hideki Omori's contemporary retirement from Tokyo collage of technology and in honor of his primary contributions to those components.
Designing fair curves and surfaces: shape quality in geometric modeling and computer-aided design
This state of the art examine of the concepts used for designing curves and surfaces for computer-aided layout functions specializes in the primary that reasonable shapes are regularly freed from unessential positive aspects and are basic in layout. The authors outline equity mathematically, show how newly built curve and floor schemes warrantly equity, and support the person in making a choice on and removal form aberrations in a floor version with no destroying the central form features of the version.
Professor Peter Hilton is without doubt one of the most sensible recognized mathematicians of his new release. He has released nearly three hundred books and papers on a variety of features of topology and algebra. the current quantity is to rejoice the party of his 60th birthday. It starts off with a bibliography of his paintings, via experiences of his contributions to topology and algebra.
Extra info for Geometric Aspects of the Abelian Modular Functions of Genus Four (I)
Sample text
X = a2 t3 + a6 t2 + a10 t + a14 . ∂u u=0 ❙✉r❢❛❝❡ ♣❛t❝❤❡s ✹✸ ❆♥② ♦t❤❡r ♣❛t❝❤ ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❜♦✉♥❞❛r② ❝✉r✈❡ ❛♥❞ ❞❡r✐✈❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ ✷ ❛t ✐ts ❡❞❣❡ ✇✐❧❧ ♠❛t❝❤ t❤✐s ♣❛t❝❤ ❛t ✐ts ✸ u = 0 ❡❞❣❡❀ s✐♠✐❧❛r ❝♦♥str❛✐♥ts ❛♣♣❧② ❛t t❤❡ ♦t❤❡r ❡❞❣❡s ✳ ■♥ ❛ ❝♦♠♠♦♥ ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛ ♥❡t✇♦r❦ ♦❢ s♣❛❝❡ ❝✉r✈❡s r❡❛❞②✲ ❞❡s✐❣♥❡❞✳ ❆♥♥♦②✐♥❣❧②✱ ✐t ✇♦r❦s ♦✉t t❤❛t ❜✐❝✉❜✐❝ ♣❛t❝❤❡s ❤❛✈❡ ❥✉st ♦♥❡ t♦♦ ♠❛♥② ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✭✐♥ ❡❛❝❤ ❞✐♠❡♥s✐♦♥✮ t♦ s✉r❢❛❝❡ s✉❝❤ ❛ ♥❡t✇♦r❦ ✇✐t❤♦✉t t❤❡ s✉♣♣❧② ♦❢ ❛❞❞✐t✐♦♥❛❧ ❞❛t❛✳ ✭❍✐❣❤❡r✲❞❡❣r❡❡ ♣❛t❝❤❡s ❤❛✈❡ ❧♦ts ♦❢ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ q✉❛❞r❛t✐❝s ❞♦♥✬t ❤❛✈❡ ❡♥♦✉❣❤✳✮ ■❢ t❤❡ ♣❛t❝❤❡s ❛r❡ ❜❡✐♥❣ ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❍❡r♠✐t❡ t❡❝❤✲ ♥✐q✉❡✱ ♦r ❛s ❛ ❣❡♦♠❡tr✐❝ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❛❧❧♦✇❛❜❧❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r♥❛❧ ♣♦✐♥ts ✐♥ ❛❞❥❛❝❡♥t ♣❛t❝❤❡s ✭♦r✖❧♦♦❦✐♥❣ ❛❤❡❛❞✖t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ♦❢ ❛ ❇é③✐❡r ❝♦♥tr♦❧ ♠❡s❤✮✱ t❤❡♥ t❤❡ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❡♠❡r❣❡ ❛s s♦✲❝❛❧❧❡❞ t✇✐st ✈❡❝t♦rs ❛t t❤❡ ♣❛t❝❤ ❝♦r♥❡rs✿ ∂ 2 Q(t, u) .
X = a2 t3 + a6 t2 + a10 t + a14 . ∂u u=0 ❙✉r❢❛❝❡ ♣❛t❝❤❡s ✹✸ ❆♥② ♦t❤❡r ♣❛t❝❤ ✇❤✐❝❤ ❤❛s t❤❡ s❛♠❡ ❜♦✉♥❞❛r② ❝✉r✈❡ ❛♥❞ ❞❡r✐✈❛t✐✈❡ ♣♦❧②♥♦♠✐❛❧ ✷ ❛t ✐ts ❡❞❣❡ ✇✐❧❧ ♠❛t❝❤ t❤✐s ♣❛t❝❤ ❛t ✐ts ✸ u = 0 ❡❞❣❡❀ s✐♠✐❧❛r ❝♦♥str❛✐♥ts ❛♣♣❧② ❛t t❤❡ ♦t❤❡r ❡❞❣❡s ✳ ■♥ ❛ ❝♦♠♠♦♥ ❝❛s❡✱ ✇❡ ❤❛✈❡ ❛ ♥❡t✇♦r❦ ♦❢ s♣❛❝❡ ❝✉r✈❡s r❡❛❞②✲ ❞❡s✐❣♥❡❞✳ ❆♥♥♦②✐♥❣❧②✱ ✐t ✇♦r❦s ♦✉t t❤❛t ❜✐❝✉❜✐❝ ♣❛t❝❤❡s ❤❛✈❡ ❥✉st ♦♥❡ t♦♦ ♠❛♥② ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ✭✐♥ ❡❛❝❤ ❞✐♠❡♥s✐♦♥✮ t♦ s✉r❢❛❝❡ s✉❝❤ ❛ ♥❡t✇♦r❦ ✇✐t❤♦✉t t❤❡ s✉♣♣❧② ♦❢ ❛❞❞✐t✐♦♥❛❧ ❞❛t❛✳ ✭❍✐❣❤❡r✲❞❡❣r❡❡ ♣❛t❝❤❡s ❤❛✈❡ ❧♦ts ♦❢ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠✱ q✉❛❞r❛t✐❝s ❞♦♥✬t ❤❛✈❡ ❡♥♦✉❣❤✳✮ ■❢ t❤❡ ♣❛t❝❤❡s ❛r❡ ❜❡✐♥❣ ❞❡t❡r♠✐♥❡❞ ❜② ❛ ❍❡r♠✐t❡ t❡❝❤✲ ♥✐q✉❡✱ ♦r ❛s ❛ ❣❡♦♠❡tr✐❝ r❡❧❛t✐♦♥s❤✐♣ ❜❡t✇❡❡♥ t❤❡ ❛❧❧♦✇❛❜❧❡ ♣♦s✐t✐♦♥s ♦❢ t❤❡ ✐♥t❡r♥❛❧ ♣♦✐♥ts ✐♥ ❛❞❥❛❝❡♥t ♣❛t❝❤❡s ✭♦r✖❧♦♦❦✐♥❣ ❛❤❡❛❞✖t❤❡ ❝♦rr❡s♣♦♥❞✐♥❣ ✈❡rt✐❝❡s ♦❢ ❛ ❇é③✐❡r ❝♦♥tr♦❧ ♠❡s❤✮✱ t❤❡♥ t❤❡ ❡①tr❛ ❞❡❣r❡❡s ♦❢ ❢r❡❡❞♦♠ ❡♠❡r❣❡ ❛s s♦✲❝❛❧❧❡❞ t✇✐st ✈❡❝t♦rs ❛t t❤❡ ♣❛t❝❤ ❝♦r♥❡rs✿ ∂ 2 Q(t, u) .
T♦ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡ ❡q✉❛t✐♦♥s✳ Hermite interpolation ■♥ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥✱ ✇❡ ❞✐✛❡r❡♥t✐❛t❡ t❤❡ ❡q✉❛t✐♦♥s ♦❢ t❤❡ ❝✉r✈❡ ♦r s✉r❢❛❝❡✱ ❛♥❞ s♦❧✈❡ s✐♠✉❧t❛♥❡♦✉s ❡q✉❛t✐♦♥s ❢♦r ❜♦t❤ ♣♦s✐t✐♦♥ ❛♥❞ t❛♥❣❡♥t ✈❛❧✉❡ ❛t ❡❛❝❤ ♦❢ t❤❡ ♣♦✐♥ts ❜❡✐♥❣ ✐♥t❡r♣♦❧❛t❡❞✳ ❚❤✉s t✇✐❝❡ ❛s ♠❛♥② ❝♦❡✣❝✐❡♥ts ❛r❡ r❡q✉✐r❡❞ ❛s ✐♥ t❤❡ ▲❛❣r❛♥❣❡ ❝❛s❡✳ ❆ ♣❛r✲ t✐❝✉❧❛r❧② ✐♠♣♦rt❛♥t ❝❛s❡ ✐s ❝♦♥str✉❝t✐♥❣ ❛ ❝✉r✈❡ ❜❡t✇❡❡♥ t✇♦ ❡♥❞ ■♥t❡r♣♦❧❛t✐♦♥ ✸✺ ✸✭✐✮✖▲❛❣r❛♥❣❡ ❛♥❞ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥ ✉s❡❞ t♦ ❝♦♥✲ str✉❝t ❛ ♣❛r❛♠❡tr✐❝ ❝✉❜✐❝ ❝✉r✈❡ s❡❣♠❡♥t✳ ♣♦✐♥ts✱ ✇✐t❤ ❦♥♦✇♥ t❛♥❣❡♥t ✈❛❧✉❡s ❛t ❡❛❝❤✳ ❚❤❛t r❡q✉✐r❡s ❛ ❝✉r✈❡ ✇✐t❤ ❢♦✉r ❝♦❡✣❝✐❡♥ts ✐♥ ❡❛❝❤ ❡q✉❛t✐♦♥✱ ✇❤✐❝❤ ❛r❡ ❝✉❜✐❝s ❀ t❤❡r❡ ✐s ❛❧s♦ ❛♥ ❡q✉✐✈❛❧❡♥t ♣❛t❝❤ ✇❤✐❝❤ r✉♥s ❜❡t✇❡❡♥ ❢♦✉r ❝♦r♥❡r ♣♦✐♥ts✱ ❛♥❞ ❤❛s ✶✻ ❝♦❡✣❝✐❡♥ts✳ ❈✉❜✐❝s ❛r❡ ❢r❡q✉❡♥t s✐❣❤t✐♥❣s ✐♥ ❝♦♠♣✉t✐♥❣ ✇✐t❤ ❣❡♦♠❡tr②✿ s❡❡ ■❧❧✉str❛t✐♦♥ ✸✭✐✮✳ The problem of parameterization ■♥t❡r♣♦❧❛t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s✱ ❞❡❝✐❞✐♥❣ ✇❤❛t t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✲ ✉❡s ❛t ❡❛❝❤ ♣♦✐♥t ✇✐❧❧ ❜❡✖t❤❡ ✐ss✉❡ ♦❢ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✖✐s ❝r✉❝✐❛❧✳ ✭❚❤❛t ✐s ❛ ♣r♦❜❧❡♠ t❤❛t ❞♦❡s ♥♦t ♦❝❝✉r ✇✐t❤ ❡①♣❧✐❝✐t✱ s✐♥❣❧❡✲✈❛❧✉❡❞✱ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s✿ ❛♥❞ s♦ ✇❡ ❝❛♥ s❡❡ ✇❤② t❤❡s❡ ❛r❡ ♣r❡❢❡rr❡❞ ❢♦r ❞r❛✇✐♥❣ ❣r❛♣❤s ❛♥❞ s♦ ♦♥✳ ❆♥❞ t❡❝❤♥✐q✉❡s ❢r♦♠ ❵❣r❛♣❤✐♥❣✬ ❛♣♣❧✐❝❛✲ t✐♦♥s ✉s✉❛❧❧② ❡①♣❧♦✐t t❤✐s ❧✐♠✐t❛t✐♦♥✱ ✇❤✐❝❤ ✐s ✇❤② ✇❡ s❤♦✉❧❞ ❜❡ ✇❛r② ♦❢ tr②✐♥❣ t♦ tr❛♥s♣❧❛♥t t❤❡♠ t♦ ♠♦r❡ ❣❡♥❡r❛❧ ❣❡♦♠❡tr✐❝ ♣r♦❜❧❡♠s✳✮ ❙♦✱ t❤❡ ♣r♦❜❧❡♠ ✇✐t❤ ✐♥t❡r♣♦❧❛t✐♥❣ ♣❛r❛♠❡tr✐❝ ❝✉r✈❡s ✐s t❤❛t✱ ✇❤✐❧❡ t❤❡ ♣♦s✐t✐♦♥s ❛♥❞ t❛♥❣❡♥t ❞✐r❡❝t✐♦♥s ♠❛② ❜❡ ♣r♦✈✐❞❡❞✱ ✇❡ ❤❛✈❡ t♦ ❡st✐♠❛t❡ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s t❤❛t t❤❡ ❝✉r✈❡ ❵s❤♦✉❧❞✬ ❤❛✈❡ ✇❤❡♥ ✐t ♣❛ss❡s ❡❛❝❤ ♣♦✐♥t✱ ❛♥❞ t❤❡ ♠❛❣♥✐t✉❞❡ ❛s ✇❡❧❧ ❛s ❞✐r❡❝t✐♦♥ ♦❢ ❞❡r✐✈❛✲ t✐✈❡s✳ ■♥ t❤❡ ❝❛s❡ ♦❢ ▲❛❣r❛♥❣❡ ✐♥t❡r♣♦❧❛t✐♦♥✱ t❤❡ s✐♠♣❧❡st ❝❤♦✐❝❡ ✐s t♦ s♣❛❝❡ ♣❛r❛♠❡tr✐❝ ✈❛❧✉❡s ❡q✉❛❧❧② ❜❡t✇❡❡♥ ♣♦✐♥t ❞❛t❛✳ ❚❤✐s ✇♦r❦s ✐❢ t❤❡ ♣♦✐♥ts ❛r❡ t❤❡♠s❡❧✈❡s q✉✐t❡ ❡✈❡♥❧② s♣❛❝❡❞❀ ♦t❤❡r✇✐s❡ s♦♠❡✲ t❤✐♥❣ ❜❡tt❡r ✐s ♥❡❡❞❡❞✳ ❙✐♥❝❡ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ✐s r❡❧❛t❡❞ t♦ ❝✉r✈❡ ❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✻ ❧❡♥❣t❤✱ ✇❡ ✇♦✉❧❞ ❧✐❦❡ t♦ ❦♥♦✇ ✇❤❛t t❤❡ ❧❡♥❣t❤ ♦❢ t❤❡ ❝✉r✈❡ ✇✐❧❧ ❜❡ ❜❡t✇❡❡♥ ❡❛❝❤ ❞❛t❛ ♣♦✐♥t❀ ❜✉t t❤❛t ✐s ♣✉tt✐♥❣ t❤❡ ❝❛rt ❜❡❢♦r❡ t❤❡ ❤♦rs❡✱ ❜❡❝❛✉s❡ ✇❡ ❤❛✈❡♥✬t ❣♦t t❤❡ ❝✉r✈❡ ②❡t✳ ❖♥❡ ❝♦✉❧❞ ✐♠♣❧❡♠❡♥t ❛ t❡❝❤♥✐q✉❡ ♦❢ s✉❝❝❡ss✐✈❡ r❡✜♥❡♠❡♥t✖s❡t ✉♣ ♦♥❡ ❝✉r✈❡✱ ❣❡t t❤❡ ❝✉r✈❡ ❧❡♥❣t❤s ❢r♦♠ ✐t✱ ❛♥❞ t❤✉s ♦❜t❛✐♥ ♥❡✇ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ❞❛t❛ ♣♦✐♥ts✱ ❛♥❞ r❡♣❡❛t t❤❡ ❡①❡r❝✐s❡✖❜✉t t❤✐s r✐❣♠❛r♦❧❡ ✐s ♥♦t ✉s✉❛❧❧② ❛t✲ t❡♠♣t❡❞❀ ✐t ✇♦✉❧❞ ♣r♦❜❛❜❧② ❜❡ ❞✐✣❝✉❧t ❡✈❡♥ t♦ ♣r♦✈❡ t❤❛t ✐t ✇♦✉❧❞ ❝♦♥✈❡r❣❡✳ ❚❤❡ ✉s✉❛❧ s♦❧✉t✐♦♥ ✐s ❝❤♦r❞✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✐③❛t✐♦♥✱ ✇❤❡r❡ t❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ♣♦✐♥ts ❛r❡ ❜❛s❡❞ ♦♥ t❤❡ ❧❡♥❣t❤s ♦❢ t❤❡ str❛✐❣❤t✲❧✐♥❡ s❡❣♠❡♥ts ❝♦♥♥❡❝t✐♥❣ t❤❡♠✳ ❚❤✐s ✐s ❛ ❣♦♦❞ ✇♦r❦❤♦rs❡✱ ❣✐✈✐♥❣ tr♦✉❜❧❡ ♦♥❧② ✇❤❡♥ t❤❡r❡ ❛r❡ ❛❜r✉♣t ❵❝♦r♥❡rs✬ ✐♠♣❧✐❡❞ ❜② t❤❡ ❞❛t❛✱ ❛♥❞ ❝❤❛♥❣❡s ♦❢ s♣❛❝✐♥❣✳ ❋✉rt❤❡r r❡✜♥❡♠❡♥ts ✐♥✈♦❧✈❡ t❛❦✐♥❣ t❤❡ ❛♥❣❧❡ ❜❡t✇❡❡♥ s✉❝❝❡ss✐✈❡ s♣❛♥s ✐♥t♦ ❛❝❝♦✉♥t ✭s❡❡ ❋❛r✐♥✬s ❜♦♦❦ ❈✉r✈❡s ❛♥❞ ❙✉r❢❛❝❡s ❢♦r ❈♦♠♣✉t❡r ❆✐❞❡❞ ●❡♦♠❡tr✐❝ ❉❡s✐❣♥ ❢♦r ♠♦r❡ ❞❡t❛✐❧✮✳ ❲✐t❤ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥✱ s✐♠✐❧❛r ♣r♦❜❧❡♠s ♦❝❝✉r❀ ❛♥❞ ✐t ♠✉st ❜❡ r❡♠❡♠❜❡r❡❞ t❤❛t ♠❛❣♥✐t✉❞❡s ♦❢ ❞❡r✐✈❛t✐✈❡s ♦❢ t❤❡ ❢♦r♠ dx dt ❡t❝✳✱ ❛r❡ r❡❧❛t❡❞ t♦ t❤❡ ❛❝t✉❛❧ s✐③❡ ♦❢ t❤❡ ❝✉r✈❡ ✐♥ t❤❡ ✉♥✐ts ♦❢ ❧❡♥❣t❤ ❜❡✐♥❣ ✉s❡❞✳ ❚❤✉s✱ ✐❢ ✇❡ s❝❛❧❡ ❛ ❝✉r✈❡ ❜② s❝❛❧✐♥❣ t❤❡ ✈❛❧✉❡s ♦❢ ✐ts ❍❡r♠✐t❡ ❝♦❡✣❝✐❡♥ts✱ ✇❡ ♠✉st s❝❛❧❡ t❤❡ ❞❡r✐✈❛t✐✈❡s ❡①♣❧✐❝✐t❧②✳ ❚❤❛t✬s ❡❛s② ❡♥♦✉❣❤ ❢♦r ❛ s✐♠♣❧❡ s❝❛❧✐♥❣✱ ❜✉t ✇❤❛t ❛❜♦✉t ❛ s❤❡❛r tr❛♥s❢♦r♠❄ ❆❧❧ t❤❡s❡ r❡♠❛r❦s ❤❛✈❡ ❜❡❡♥ ❛❞❞r❡ss❡❞ t♦ t❤❡ ♣r♦❜❧❡♠ ♦❢ ✐♥t❡r✲ ♣♦❧❛t✐♦♥✱ ❜✉t ❛❧s♦ ❛♣♣❧② t♦ ❝✉r✈❡ ✜tt✐♥❣✳ ❆❣❛✐♥✱ t❤✐s ✐s ❛ ♣r♦❝❡ss t❤❛t ✇♦r❦s ✇❡❧❧ ✇✐t❤ ❡①♣❧✐❝✐t ❣❡♦♠❡tr②✱ ❛♥❞ ❢❛✐r❧② ✇❡❧❧ ✇✐t❤ ✐♠♣❧✐❝✲ ✐ts ✭❡①❝❡♣t t❤❛t ♥♦r♠❛❧✐③❛t✐♦♥ ❝❛✉s❡s ❛ ♣r♦❜❧❡♠✮✳ ❲✐t❤ ♣❛r❛♠❡tr✐❝ ❣❡♦♠❡tr②✱ ✇❡ ❛❣❛✐♥ ❤❛✈❡ t♦ ❞❡❝✐❞❡ ✐♥ ❛❞✈❛♥❝❡ ✇❤❛t ♣❛r❛♠❡t❡r ✈❛❧✉❡ ❡❛❝❤ ♣♦✐♥t ✇✐❧❧ ❝♦rr❡s♣♦♥❞ t♦✳ ❇✉t ✐❢ t❤❡ ♣♦✐♥ts ❛r❡ ❛t ❛❧❧ ❞❡♥s❡✱ t❤✐s ✐s ❞✐✣❝✉❧t✿ ❝❤♦r❞✲❧❡♥❣t❤ ♣❛r❛♠❡t❡r✐③❛t✐♦♥ ✐s ❝❡rt❛✐♥❧② ✉s❡❧❡ss✳ ❲❡ ❝♦♥❝❧✉❞❡ t❤✐s s❡❝t✐♦♥ ✇✐t❤ ❈ ❝♦❞❡ ❢♦r ▲❛❣r❛♥❣❡ ❛♥❞ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛t✐♦♥✳ ❚❤❡ ✜rst ♣r♦❝❡❞✉r❡ ✇♦r❦s ♦✉t t❤❡ ▲❛❣r❛♥❣✐❛♥ ✐♥t❡r✲ ♣♦❧❛t✐♥❣ ❝✉❜✐❝ ♣❛r❛♠❡tr✐❝ ♣♦❧②♥♦♠✐❛❧ t❤r♦✉❣❤ ❢♦✉r ♣♦✐♥ts ✐♥ t❤r❡❡ ❞✐♠❡♥s✐♦♥s✳ ❚❤❡ ♣♦✐♥ts ✇✐❧❧ ❜❡ s✉♣♣❧✐❡❞ ✐♥ ♣①✱ ♣②✱ ❛♥❞ ♣③✳ ❚❤❡ 0✱ ❛♥❞ t❤❡ ♣❛r❛♠✲ ♣❛r❛♠❡t❡r ♦♥ t❤❡ ❝✉r✈❡ ❛t t❤❡ ✜rst ♣♦✐♥t ✇✐❧❧ ❜❡ 1❀ t❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ♣♦❧②♥♦♠✐❛❧ ✇✐❧❧ ❜❡ ♣♦❧②①✱ ♣♦❧②②✱ ❛♥❞ ♣♦❧②③❀ ♣♦❧②①❬✸❪ ✐s t❤❡ ❝♦❡✣❝✐❡♥t ♦❢ ❡t❡r ❛t t❤❡ ❧❛st ♣♦✐♥t r❡t✉r♥❡❞ ✐♥ ■♥t❡r♣♦❧❛t✐♦♥ t3 ✐♥ x ✸✼ ❛♥❞ s♦ ♦♥✳ ❚❤❡ ♣❛r❛♠❡t❡r ✈❛❧✉❡s ❛t t❤❡ ♠✐❞❞❧❡ t✇♦ ♣♦✐♥ts ♦♥ t❤❡ ❝✉r✈❡ ✇✐❧❧ ❜❡ r❡t✉r♥❡❞ ✐♥ t✶ ❛♥❞ t✷✳ ★✐♥❝❧✉❞❡ ❁♠❛t❤✳❤❃ ★✐♥❝❧✉❞❡ ❁st❞✐♦✳❤❃ ✴✯ ❆❜s♦❧✉t❡ ✈❛❧✉❡ ♠❛❝r♦ ✯✴ ★❞❡❢✐♥❡ ❢❛❜s✭❛✮ ✭✭✭❛✮ ❁ ✵✳✵✮ ❄ ✭✲✭❛✮✮ ✿ ✭❛✮✮ ✴✯ ❆❧♠♦st ✵ ✲ ❛❞❥✉st ❢♦r ②♦✉r ❛♣♣❧✐❝❛t✐♦♥ ✯✴ ★❞❡❢✐♥❡ ❆❈❈❨ ✭✶✳✵❡✲✻✮ ✐♥t ❧❛❣r❛♥❣❡✭♣①✱♣②✱♣③✱t✶✱t✷✱♣♦❧②①✱♣♦❧②②✱♣♦❧②③✮ ❢❧♦❛t ♣①❬✹❪✱♣♦❧②①❬✹❪❀ ❢❧♦❛t ♣②❬✹❪✱♣♦❧②②❬✹❪❀ ❢❧♦❛t ♣③❬✹❪✱♣♦❧②③❬✹❪❀ ❢❧♦❛t ✯t✶✱✯t✷❀ ④ ❢❧♦❛t ①❞✱②❞✱③❞✱❞❢❧❀ ✐♥t ✐❀ ✴✯ ✯✴ ❙✉♠ t❤❡ ❞✐st❛♥❝❡s ❜❡t✇❡❡♥ t❤❡ ♣♦✐♥ts t♦ ✉s❡ t♦ s❝❛❧❡ t✶ ❛♥❞ t✷✳ ◆♦t❡ t❤❡ ❡①tr❡♠❡❧② t✐r❡s♦♠❡ ❝❛st✐♥❣ t❤❛t ♥❡❡❞s t♦ ❜❡ ❞♦♥❡ ❜❡❝❛✉s❡ ❛❧❧ ♦❢ t❤❡ st❛♥❞❛r❞ ❈ ♠❛t❤s ❧✐❜r❛r② ✐s ✐♥ ❞♦✉❜❧❡s✳ ❞❢❧ ❂ ✵✳✵❀ ❢♦r✭✐ ❂ ✶❀ ✐ ❁ ✹❀ ✐✰✰✮ ④ ①❞ ❂ ♣①❬✐❪ ✲ ♣①❬✐✲✶❪❀ ②❞ ❂ ♣②❬✐❪ ✲ ♣②❬✐✲✶❪❀ ③❞ ❂ ♣③❬✐❪ ✲ ♣③❬✐✲✶❪❀ ❞❢❧ ❂ ❞❢❧ ✰ ✭❢❧♦❛t✮sqrt✭✭❞♦✉❜❧❡✮ ✭①❞✯①❞ ✰ ②❞✯②❞ ✰ ③❞✯③❞✮✮❀ ❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✸✽ ⑥ ✐❢ ✭✐ ❂❂ ✶✮ ✯t✶ ❂ ❞❢❧❀ ✐❢ ✭✐ ❂❂ ✷✮ ✯t✷ ❂ ❞❢❧❀ ✐❢ ✭❞❢❧ ❁ ❆❈❈❨✮ ④ ❢♣r✐♥t❢✭st❞❡rr✱ ✧▲❛❣r❛♥❣❡✿ ❝✉r✈❡ t♦♦ s❤♦rt✿ ✪❢❭♥✧✱❞❢❧✮❀ r❡t✉r♥✭✶✮❀ ⑥ ✯t✶ ❂ ✯t✶✴❞❢❧❀ ✯t✷ ❂ ✯t✷✴❞❢❧❀ ✴✯ ✯✴ ❈❛❧❧ t❤❡ ♣r♦❝❡❞✉r❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡❢❢✐❝✐❡♥ts ✐♥ ❡❛❝❤ ❝♦♦r❞✐♥❛t❡✳ ✐❢✭❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣①✱♣♦❧②①✱t✶✱t✷✮✮ r❡t✉r♥✭✷✮❀ ✐❢✭❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣②✱♣♦❧②②✱t✶✱t✷✮✮ r❡t✉r♥✭✸✮❀ ✐❢✭❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣③✱♣♦❧②③✱t✶✱t✷✮✮ r❡t✉r♥✭✹✮❀ r❡t✉r♥✭✵✮❀ ⑥ ✴✯ ❧❛❣r❛♥❣❡ ✯✴ ✴✯ ✯✴ r♦❝❡❞✉r❡ t♦ ❝♦♠♣✉t❡ t❤❡ ❝♦❡❢❢✐❝✐❡♥ts ✐♥ ♦♥❡ ❞✐♠❡♥s✐♦♥ ♦❢ t❤❡ ▲❛❣r❛♥❣✐❛♥ ❝✉❜✐❝ t❤r♦✉❣❤ ❢♦✉r ♣♦✐♥ts✳ ❚❤❡ ❝♦❞❡ r❡❢❧❡❝ts t❤❡ ❛❧❣❡❜r❛✳ ✐♥t ❧❛❣r❛♥❣❡❴❝♦❡❢❢s✭♣✱♣♦❧②✱t✶✱t✷✮ ❢❧♦❛t ♣❬✹❪✱♣♦❧②❬✹❪❀ ❢❧♦❛t ✯t✶✱✯t✷❀ ④ ❢❧♦❛t ❞✶✱❞✷✱❞✸✱t✶s✱t✷s✱t✶❝✱t✷❝✱❞❡♥♦♠✱tt❀ ■♥t❡r♣♦❧❛t✐♦♥ ✸✾ ❞✶ ❂ ♣❬✶❪ ✲ ♣❬✵❪❀ ❞✷ ❂ ♣❬✷❪ ✲ ♣❬✵❪❀ ❞✸ ❂ ♣❬✸❪ ✲ ♣❬✵❪❀ t✶s ❂ ✭✯t✶✮✯✭✯t✶✮❀ t✷s ❂ ✭✯t✷✮✯✭✯t✷✮❀ t✶❝ ❂ t✶s✯✭✯t✶✮❀ t✷❝ ❂ t✷s✯✭✯t✷✮❀ ❞❡♥♦♠ ❂ ✭t✷s ✲ ✭✯t✷✮✮✯t✶❝❀ ✐❢ ✭❢❛❜s✭❞❡♥♦♠✮ ❁ ❆❈❈❨✮ ④ ❢♣r✐♥t❢✭st❞❡rr✱ ✧▲❛❣r❛♥❣❡❴❝♦❡❢❢s✿ ✐♥❝r❡♠❡♥ts t♦♦ s❤♦rt✿ ✪❢❭♥✧✱ ❞❡♥♦♠✮❀ r❡t✉r♥✭✶✮❀ ⑥ tt ❂ ✭✲t✷❝ ✰ ✭✯t✷✮✮✯t✶s ✰ ✭t✷❝ ✲ t✷s✮✯✭✯t✶✮❀ ♣♦❧②❬✸❪ ❂ ✭❞✸✯✭✯t✷✮ ✲ ❞✷✮✯t✶s ✰ ✭✲❞✸✯t✷s ✰ ❞✷✮✯✭✯t✶✮ ✰ ❞✶✯t✷s ✲ ❞✶✯✭✯t✷✮✴❞❡♥♦♠ ✰ tt❀ ♣♦❧②❬✷❪ ❂ ✭✲❞✸✯✭✯t✷✮ ✰ ❞✷✮✯t✶❝ ✰ ✭❞✸✯t✷❝ ✲ ❞✷✮✯✭✯t✶✮ ✰ ❞✶✯t✷❝ ✰ ❞✶✯✭✯t✷✮✴❞❡♥♦♠ ✰ tt❀ ♣♦❧②❬✶❪ ❂ ✭❞✸✯t✷s ✲ ❞✷✮✯t✶❝ ✰ ✭✲❞✸✯t✷❝ ✰ ❞✷✮✯t✶s ✰ ❞✶✯t✷❝ ✲ ❞✶✯t✷s✴❞❡♥♦♠ ✰ tt❀ ♣♦❧②❬✵❪ ❂ ♣❬✵❪❀ r❡t✉r♥✭✵✮❀ ⑥ ✴✯ ❧❛❣r❛♥❣❡❴❝♦❡❢❢s ✯✴ ❚❤❡ s❡❝♦♥❞ ♣r♦❝❡❞✉r❡ ❝♦♠♣✉t❡s t❤❡ ❍❡r♠✐t❡ ✐♥t❡r♣♦❧❛♥t ✐♥ t❤r❡❡ ❛r❛♠❡tr✐❝ ❝✉r✈❡s ❛♥❞ s✉r❢❛❝❡s ✹✵ ❞✐♠❡♥s✐♦♥s t❤r♦✉❣❤ t✇♦ ♣♦✐♥ts ✇✐t❤ t✇♦ ❣r❛❞✐❡♥t ✈❡❝t♦rs ❛t t❤❡ ❡♥❞s✳ ❚❤❡ ♣♦✐♥ts ❛r❡ ♣✵ ❛♥❞ ♣✶✱ ❛♥❞ t❤❡ ❣r❛❞✐❡♥ts ❛r❡ ❣✵ ❛♥❞ ❣✶✳ ❚❤❡ ❝♦❡✣❝✐❡♥ts ♦❢ t❤❡ ✐♥t❡r♣♦❧❛t✐♥❣ ♣♦❧②♥♦♠✐❛❧ ❛r❡ r❡t✉r♥❡❞ ✐♥ ♣♦❧②①✱ ♣♦❧②②✱ ❛♥❞ ♣♦❧②③✳ ❚❤❡ ❛❧❣❡❜r❛ ✐♥ t❤✐s ❝❛s❡ ✐s ♠✉❝❤ s✐♠♣❧❡r t❤❛♥ t❤❛t ❢♦r ▲❛❣r❛♥❣✐❛♥ ✐♥t❡r♣♦❧❛t✐♦♥✳ ✈♦✐❞ ❤❡r♠✐t❡✭♣✵✱♣✶✱❣✵✱❣✶✱♣♦❧②①✱♣♦❧②②✱♣♦❧②③✮ ❢❧♦❛t ♣✵❬✸❪✱♣✶❬✸❪✱❣✵❬✸❪✱❣✶❬✸❪❀ ❢❧♦❛t ♣♦❧②①❬✹❪✱♣♦❧②②❬✹❪✱♣♦❧②③❬✹❪❀ ④ ✈♦✐❞ ❤❡r♠✐t❡❴❝♦❡❢❢s✭✮❀ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵❬✵❪✱♣✶❬✵❪✱❣✵❬✵❪✱❣✶❬✵❪✱♣♦❧②①✮❀ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵❬✶❪✱♣✶❬✶❪✱❣✵❬✶❪✱❣✶❬✶❪✱♣♦❧②②✮❀ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵❬✷❪✱♣✶❬✷❪✱❣✵❬✷❪✱❣✶❬✷❪✱♣♦❧②③✮❀ ⑥ ✴✯ ❤❡r♠✐t❡ ✯✴ ✈♦✐❞ ❤❡r♠✐t❡❴❝♦❡❢❢s✭♣✵✱♣✶✱❣✵✱❣✶✱♣♦❧②✮ ❢❧♦❛t ♣✵✱♣✶✱❣✵✱❣✶❀ ❢❧♦❛t ♣♦❧②❬✹❪❀ ④ ❢❧♦❛t ❞✱❣❀ ❞ ❂ ♣✶ ✲ ♣✵ ✲ ❣✵❀ ❣ ❂ ❣✶ ✲ ❣✵❀ ♣♦❧②❬✵❪ ♣♦❧②❬✶❪ ♣♦❧②❬✷❪ ♣♦❧②❬✸❪ ❂ ❂ ❂ ❂ ♣✵❀ ❣✵❀ ✸✳✵✯❞ ✲ ❣❀ ✲✷✳✵✯❞ ✰ ❣❀ ⑥ ✴✯ ❤❡r♠✐t❡❴❝♦❡❢❢s ✯✴ Surface patches ❙✉r❢❛❝❡ ♣❛t❝❤❡s ✹✶ Q = F(t, u) 0 ≤ t ≤ 1, 0 ≤ u ≤ 1✳ ✸✭✐✐✮✖❆ ♣❛r❛♠❡tr✐❝ ♣❛t❝❤ ✐♥t❡r✈❛❧ ❞❡✜♥❡❞ ♦✈❡r t❤❡ ❙✉r❢❛❝❡ ♣❛t❝❤❡s ❛r❡ ♣❛r❛♠❡tr✐❝ s✉r❢❛❝❡s ♦❢ t❤❡ ❢♦r♠ x = f1 (t, u) y = f2 (t, u) z = f3 (t, u) ✭✇❤✐❝❤ ✇❡ ❝❛♥ ❛❧s♦ ✇r✐t❡ ✇✐t❤ ✈❡❝t♦r ❝♦❡✣❝✐❡♥ts ✱ ❛s Q = F(t, u) ❛ ♣❛♣❡r✲s❛✈✐♥❣ ♠❡❛s✉r❡ t❤❛t ✇✐❧❧ ❜❡ ✐♥❝r❡❛s✐♥❣❧② ✉s❡❞ ✐♥ t❤✐s ❝❤❛♣t❡r✮✳ ❆ ♣❛t❝❤ ❝❛♥ ❜❡ ❞❡✜♥❡❞ ♦✈❡r ❛♥② ♣❛r❛♠❡tr✐❝ ♣♦rt✐♦♥ ♦❢ t❤❡ (t, u) ♣❛r❛♠❡t❡r s♣❛❝❡ ✱ ❜✉t ✐s ❡❛s✐❡st t♦ ❞❡❛❧ ✇✐t❤ ♦✈❡r t❤❡ t✇♦✲❞✐♠❡♥s✐♦♥❛❧ ✐♥t❡r✈❛❧✿ 0≤ t ≤1 0 ≤ u ≤ 1.