By Adrian Bowyer
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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.