s tt t s t rs - - PowerPoint PPT Presentation

▼❛♣s t❤❛t t❛❦❡ ❧✐♥❡s t♦ ♣❧❛♥❡ ❝✉r✈❡s

❱s❡✈♦❧♦❞ P❡tr✉s❝❤❡♥❦♦✱ ❱❧❛❞❧❡♥ ❚✐♠♦r✐♥∗

∗❋❛❝✉❧t② ♦❢ ▼❛t❤❡♠❛t✐❝s

◆❛t✐♦♥❛❧ ❘❡s❡❛r❝❤ ❯♥✐✈❡rs✐t② ❍✐❣❤❡r ❙❝❤♦♦❧ ♦❢ ❊❝♦♥♦♠✐❝s

❋✐❡❧❞s ■♥st✐t✉t❡✱ ❚♦r♦♥t♦✱ ◆♦✈❡♠❜❡r ✷✽✱ ✷✵✶✹

P❧❛♥❛r✐③❛t✐♦♥s

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ ✐s ❛ s✉✣❝✐❡♥t❧② s♠♦♦t❤ ♠❛♣♣✐♥❣ f : U ⊂ RP2 → RP3 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ RP2✱ t❤❡ s❡t f (U ∩ L) ✐s ♣❧❛♥❛r✳

❉❡✜♥✐t✐♦♥

❚✇♦ ♣❧❛♥❛r✐③❛t✐♦♥s f : U → RP3 ❛♥❞ g : V → RP3 ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t W ⊂ U ∩ V s✉❝❤ t❤❛t f = g ♦♥ W ✱ ✉♣ t♦ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t s♣❛❝❡s✳

Pr♦❜❧❡♠

❈❧❛ss✐❢② ♣❧❛♥❛r✐③❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

P❧❛♥❛r✐③❛t✐♦♥s

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ ✐s ❛ s✉✣❝✐❡♥t❧② s♠♦♦t❤ ♠❛♣♣✐♥❣ f : U ⊂ RP2 → RP3 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ RP2✱ t❤❡ s❡t f (U ∩ L) ✐s ♣❧❛♥❛r✳

❉❡✜♥✐t✐♦♥

❚✇♦ ♣❧❛♥❛r✐③❛t✐♦♥s f : U → RP3 ❛♥❞ g : V → RP3 ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t W ⊂ U ∩ V s✉❝❤ t❤❛t f = g ♦♥ W ✱ ✉♣ t♦ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t s♣❛❝❡s✳

Pr♦❜❧❡♠

❈❧❛ss✐❢② ♣❧❛♥❛r✐③❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

P❧❛♥❛r✐③❛t✐♦♥s

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ ✐s ❛ s✉✣❝✐❡♥t❧② s♠♦♦t❤ ♠❛♣♣✐♥❣ f : U ⊂ RP2 → RP3 s✉❝❤ t❤❛t✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ RP2✱ t❤❡ s❡t f (U ∩ L) ✐s ♣❧❛♥❛r✳

❉❡✜♥✐t✐♦♥

❚✇♦ ♣❧❛♥❛r✐③❛t✐♦♥s f : U → RP3 ❛♥❞ g : V → RP3 ❛r❡ ❡q✉✐✈❛❧❡♥t ✐❢ t❤❡r❡ ✐s ❛ ♥♦♥❡♠♣t② ♦♣❡♥ s✉❜s❡t W ⊂ U ∩ V s✉❝❤ t❤❛t f = g ♦♥ W ✱ ✉♣ t♦ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥s ♦❢ t❤❡ s♦✉r❝❡ ❛♥❞ t❛r❣❡t s♣❛❝❡s✳

Pr♦❜❧❡♠

❈❧❛ss✐❢② ♣❧❛♥❛r✐③❛t✐♦♥s ❛❝❝♦r❞✐♥❣ t♦ t❤✐s ❡q✉✐✈❛❧❡♥❝❡ r❡❧❛t✐♦♥✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr②

❚❤❡♦r❡♠ ✭▼☎ ♦❜✐✉s✱ ✶✽✷✼✮

❙✉♣♣♦s❡ t❤❛t f : RPn → RPn ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥❡✲t♦✲♦♥❡ ♠❛♣ t❛❦✐♥❣ ❛❧❧ str❛✐❣❤t ❧✐♥❡s t♦ str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡♥ f ✐s ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐✳❡✳✱ ❛ ♣r♦❥❡❝t✐✈✐③❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ Rn+1 → Rn+1✳

❚❤❡♦r❡♠ ✭✈♦♥ ❙t❛✉❞t✮

❚❤❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥ ✐s s✉♣❡r✢✉♦✉s✳

❘❡♠❛r❦

❚❤✐s t❤❡♦r❡♠ ❤❛s ❧♦❝❛❧ ✈❡rs✐♦♥s✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr②

❚❤❡♦r❡♠ ✭▼☎ ♦❜✐✉s✱ ✶✽✷✼✮

❙✉♣♣♦s❡ t❤❛t f : RPn → RPn ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥❡✲t♦✲♦♥❡ ♠❛♣ t❛❦✐♥❣ ❛❧❧ str❛✐❣❤t ❧✐♥❡s t♦ str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡♥ f ✐s ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐✳❡✳✱ ❛ ♣r♦❥❡❝t✐✈✐③❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ Rn+1 → Rn+1✳

❚❤❡♦r❡♠ ✭✈♦♥ ❙t❛✉❞t✮

❚❤❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥ ✐s s✉♣❡r✢✉♦✉s✳

❘❡♠❛r❦

❚❤✐s t❤❡♦r❡♠ ❤❛s ❧♦❝❛❧ ✈❡rs✐♦♥s✳

❚❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡ ●❡♦♠❡tr②

❚❤❡♦r❡♠ ✭▼☎ ♦❜✐✉s✱ ✶✽✷✼✮

❙✉♣♣♦s❡ t❤❛t f : RPn → RPn ✐s ❛ ❝♦♥t✐♥✉♦✉s ♦♥❡✲t♦✲♦♥❡ ♠❛♣ t❛❦✐♥❣ ❛❧❧ str❛✐❣❤t ❧✐♥❡s t♦ str❛✐❣❤t ❧✐♥❡s✳ ❚❤❡♥ f ✐s ❛ ♣r♦❥❡❝t✐✈❡ tr❛♥s❢♦r♠❛t✐♦♥✱ ✐✳❡✳✱ ❛ ♣r♦❥❡❝t✐✈✐③❛t✐♦♥ ♦❢ ❛ ❧✐♥❡❛r ✐s♦♠♦r♣❤✐s♠ Rn+1 → Rn+1✳

❚❤❡♦r❡♠ ✭✈♦♥ ❙t❛✉❞t✮

❚❤❡ ❝♦♥t✐♥✉✐t② ❛ss✉♠♣t✐♦♥ ✐s s✉♣❡r✢✉♦✉s✳

❘❡♠❛r❦

❚❤✐s t❤❡♦r❡♠ ❤❛s ❧♦❝❛❧ ✈❡rs✐♦♥s✳

❈❧❛ss✐❝❛❧ ❣❡♦♠❡t❡rs

❆✉❣✉st ▼☎ ♦❜✐✉s ✶✼✾✵✕✶✽✻✽ ❑❛r❧ ●❡♦r❣ ❈❤r✐st✐❛♥ ✈♦♥ ❙t❛✉❞t ✶✼✾✽✕✶✽✻✼

▼♦t✐✈❛t✐♦♥

• ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡
• ❡♦♠❡tr②
• ▲❡t L ❜❡ ❛ ❧✐♥❡❛r s②st❡♠ ♦❢ ❝✉r✈❡s ✭❡✳❣✳✱ t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ❧✐♥❡s✱

❝✐r❝❧❡s✱ ❝♦♥✐❝s✱ ❡t❝✳✮✳ ❙t✉❞②✐♥❣ ♠❛♣♣✐♥❣s f : U ⊂ RP2 → RP2 t❛❦✐♥❣ ❧✐♥❡ s❡❣♠❡♥ts t♦ ❝✉r✈❡s ❢r♦♠ L ✐s r❡❧❛t❡❞ ✇✐t❤ st✉❞②✐♥❣ ♣❧❛♥❛r✐③❛t✐♦♥s✳

▼♦t✐✈❛t✐♦♥

• ❆♥ ❡①t❡♥s✐♦♥ ♦❢ t❤❡ ❋✉♥❞❛♠❡♥t❛❧ ❚❤❡♦r❡♠ ♦❢ Pr♦❥❡❝t✐✈❡
• ❡♦♠❡tr②
• ▲❡t L ❜❡ ❛ ❧✐♥❡❛r s②st❡♠ ♦❢ ❝✉r✈❡s ✭❡✳❣✳✱ t❤❡ ❢❛♠✐❧② ♦❢ ❛❧❧ ❧✐♥❡s✱

❝✐r❝❧❡s✱ ❝♦♥✐❝s✱ ❡t❝✳✮✳ ❙t✉❞②✐♥❣ ♠❛♣♣✐♥❣s f : U ⊂ RP2 → RP2 t❛❦✐♥❣ ❧✐♥❡ s❡❣♠❡♥ts t♦ ❝✉r✈❡s ❢r♦♠ L ✐s r❡❧❛t❡❞ ✇✐t❤ st✉❞②✐♥❣ ♣❧❛♥❛r✐③❛t✐♦♥s✳

❚r✐✈✐❛❧ ❝❛s❡s

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s tr✐✈✐❛❧ ✐❢ f (U) ❧✐❡s ✐♥ ❛ ♣❧❛♥❡✳

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❝♦✲tr✐✈✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣♦✐♥t a ∈ RP3 s✉❝❤ t❤❛t f (U ∩ L) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♣❧❛♥❡ t❤r♦✉❣❤ a✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ RP2✳

❚r✐✈✐❛❧ ❝❛s❡s

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s tr✐✈✐❛❧ ✐❢ f (U) ❧✐❡s ✐♥ ❛ ♣❧❛♥❡✳

❉❡✜♥✐t✐♦♥

❆ ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❝♦✲tr✐✈✐❛❧ ✐❢ t❤❡r❡ ❡①✐sts ❛ ♣♦✐♥t a ∈ RP3 s✉❝❤ t❤❛t f (U ∩ L) ✐s ❝♦♥t❛✐♥❡❞ ✐♥ ❛ ♣❧❛♥❡ t❤r♦✉❣❤ a✱ ❢♦r ❡✈❡r② ❧✐♥❡ L ⊂ RP2✳

❈♦✲tr✐✈✐❛❧ ♣❧❛♥❛r✐③❛t✐♦♥s

◆♦♥✲tr✐✈✐❛❧ ❡①❛♠♣❧❡s

❉❡✜♥✐t✐♦♥

❆ q✉❛❞r❛t✐❝ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ ✐s ❛ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ f : RP2 RP2 ❣✐✈❡♥ ✐♥ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡s ❜② ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ✷✿ f [x0 : x1 : x2] = [y0 : y1 : y2 : y3], yα =

2

• i,j=0

ai,j

α xixj.

❊①❛♠♣❧❡

❆♥② q✉❛❞r❛t✐❝ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ ✐s ❛ ♣❧❛♥❛r✐③❛t✐♦♥❀ ✐t t❛❦❡s ❧✐♥❡s t♦ ❝♦♥✐❝s✳

◆♦♥✲tr✐✈✐❛❧ ❡①❛♠♣❧❡s

❉❡✜♥✐t✐♦♥

❆ q✉❛❞r❛t✐❝ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ ✐s ❛ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ f : RP2 RP2 ❣✐✈❡♥ ✐♥ ❤♦♠♦❣❡♥❡♦✉s ❝♦♦r❞✐♥❛t❡s ❜② ❤♦♠♦❣❡♥❡♦✉s ♣♦❧②♥♦♠✐❛❧s ♦❢ ❞❡❣r❡❡ ✷✿ f [x0 : x1 : x2] = [y0 : y1 : y2 : y3], yα =

2

• i,j=0

ai,j

α xixj.

❊①❛♠♣❧❡

❆♥② q✉❛❞r❛t✐❝ r❛t✐♦♥❛❧ ♠❛♣♣✐♥❣ ✐s ❛ ♣❧❛♥❛r✐③❛t✐♦♥❀ ✐t t❛❦❡s ❧✐♥❡s t♦ ❝♦♥✐❝s✳

❆ ❙t❡✐♥❡r s✉r❢❛❝❡

❆ ❙t❡✐♥❡r s✉r❢❛❝❡

❆ ❙t❡✐♥❡r s✉r❢❛❝❡

❉✉❛❧✐t②

• ❚❤✐s ✐s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣r♦❥❡❝t✐✈❡ ❞✉❛❧✐t② ❢♦r

♣❧❛♥❛r✐③❛t✐♦♥s✳

• ❋♦r ❡✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3✱ t❤❡r❡ ✐s t❤❡ ❞✉❛❧

♣❧❛♥❛r✐③❛t✐♦♥ f ∗ : U∗ → RP3∗✳

• ❚❤❡ ♦♣❡♥ s❡t U∗✱ ♣♦ss✐❜❧② ❡♠♣t②✱ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧

❧✐♥❡s L ∈ RP2∗ s✉❝❤ t❤❛t f (L ∩ U) ❧✐❡s ✐♥ ❛ ✉♥✐q✉❡ ♣❧❛♥❡ PL✳

• ❚❤❡ ♠❛♣ f ∗ s❡♥❞s L t♦ PL ∈ RP3∗✳

❉✉❛❧✐t②

• ❚❤✐s ✐s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣r♦❥❡❝t✐✈❡ ❞✉❛❧✐t② ❢♦r

♣❧❛♥❛r✐③❛t✐♦♥s✳

• ❋♦r ❡✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3✱ t❤❡r❡ ✐s t❤❡ ❞✉❛❧

♣❧❛♥❛r✐③❛t✐♦♥ f ∗ : U∗ → RP3∗✳

• ❚❤❡ ♦♣❡♥ s❡t U∗✱ ♣♦ss✐❜❧② ❡♠♣t②✱ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧

❧✐♥❡s L ∈ RP2∗ s✉❝❤ t❤❛t f (L ∩ U) ❧✐❡s ✐♥ ❛ ✉♥✐q✉❡ ♣❧❛♥❡ PL✳

• ❚❤❡ ♠❛♣ f ∗ s❡♥❞s L t♦ PL ∈ RP3∗✳

❉✉❛❧✐t②

• ❚❤✐s ✐s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣r♦❥❡❝t✐✈❡ ❞✉❛❧✐t② ❢♦r

♣❧❛♥❛r✐③❛t✐♦♥s✳

• ❋♦r ❡✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3✱ t❤❡r❡ ✐s t❤❡ ❞✉❛❧

♣❧❛♥❛r✐③❛t✐♦♥ f ∗ : U∗ → RP3∗✳

• ❚❤❡ ♦♣❡♥ s❡t U∗✱ ♣♦ss✐❜❧② ❡♠♣t②✱ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧

❧✐♥❡s L ∈ RP2∗ s✉❝❤ t❤❛t f (L ∩ U) ❧✐❡s ✐♥ ❛ ✉♥✐q✉❡ ♣❧❛♥❡ PL✳

• ❚❤❡ ♠❛♣ f ∗ s❡♥❞s L t♦ PL ∈ RP3∗✳

❉✉❛❧✐t②

• ❚❤✐s ✐s ❛♥ ✐♠♣❧❡♠❡♥t❛t✐♦♥ ♦❢ ♣r♦❥❡❝t✐✈❡ ❞✉❛❧✐t② ❢♦r

♣❧❛♥❛r✐③❛t✐♦♥s✳

• ❋♦r ❡✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3✱ t❤❡r❡ ✐s t❤❡ ❞✉❛❧

♣❧❛♥❛r✐③❛t✐♦♥ f ∗ : U∗ → RP3∗✳

• ❚❤❡ ♦♣❡♥ s❡t U∗✱ ♣♦ss✐❜❧② ❡♠♣t②✱ ✐s ❞❡✜♥❡❞ ❛s t❤❡ s❡t ♦❢ ❛❧❧

❧✐♥❡s L ∈ RP2∗ s✉❝❤ t❤❛t f (L ∩ U) ❧✐❡s ✐♥ ❛ ✉♥✐q✉❡ ♣❧❛♥❡ PL✳

• ❚❤❡ ♠❛♣ f ∗ s❡♥❞s L t♦ PL ∈ RP3∗✳

❚❤❡ r❡s✉❧t

❚❤❡♦r❡♠

❊✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ♣❧❛♥❛r✐③❛t✐♦♥ t❤❛t ✐s

• tr✐✈✐❛❧✱ ❖❘
• ❝♦✲tr✐✈✐❛❧✱ ❖❘
• q✉❛❞r❛t✐❝✱ ❖❘
• ❞✉❛❧ q✉❛❞r❛t✐❝✳

❚❤❡ r❡s✉❧t

❚❤❡♦r❡♠

❊✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ♣❧❛♥❛r✐③❛t✐♦♥ t❤❛t ✐s

• tr✐✈✐❛❧✱ ❖❘
• ❝♦✲tr✐✈✐❛❧✱ ❖❘
• q✉❛❞r❛t✐❝✱ ❖❘
• ❞✉❛❧ q✉❛❞r❛t✐❝✳

❚❤❡ r❡s✉❧t

❚❤❡♦r❡♠

❊✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ♣❧❛♥❛r✐③❛t✐♦♥ t❤❛t ✐s

• tr✐✈✐❛❧✱ ❖❘
• ❝♦✲tr✐✈✐❛❧✱ ❖❘
• q✉❛❞r❛t✐❝✱ ❖❘
• ❞✉❛❧ q✉❛❞r❛t✐❝✳

❚❤❡ r❡s✉❧t

❚❤❡♦r❡♠

❊✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ♣❧❛♥❛r✐③❛t✐♦♥ t❤❛t ✐s

• tr✐✈✐❛❧✱ ❖❘
• ❝♦✲tr✐✈✐❛❧✱ ❖❘
• q✉❛❞r❛t✐❝✱ ❖❘
• ❞✉❛❧ q✉❛❞r❛t✐❝✳

❚❤❡ r❡s✉❧t

❚❤❡♦r❡♠

❊✈❡r② ♣❧❛♥❛r✐③❛t✐♦♥ f : U → RP3 ✐s ❡q✉✐✈❛❧❡♥t t♦ ❛ ♣❧❛♥❛r✐③❛t✐♦♥ t❤❛t ✐s

• tr✐✈✐❛❧✱ ❖❘
• ❝♦✲tr✐✈✐❛❧✱ ❖❘
• q✉❛❞r❛t✐❝✱ ❖❘
• ❞✉❛❧ q✉❛❞r❛t✐❝✳

❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥

❚❤❡♦r❡♠

❚❤❡r❡ ❛r❡ ✶✻ ❡q✉✐✈❛❧❡♥❝❡ ❝❧❛ss❡s ♦❢ ♥♦♥✲✭❝♦✮✲tr✐✈✐❛❧ ♣❧❛♥❛r✐③❛t✐♦♥s✿ (Q1)✿ [x : y : z] → [xy : xz : yz : x2 + y2 + z2] (Q2)✿ [x : y : z] → [xy : xz : yz : x2 − y2 + z2] (Q3)✿ [x : y : z] → [x2 + y2 : y2 + z2 : xz : yz] (Q4)✿ [x : y : z] → [x2 − y2 : xy : yz : z2] (Q5)✿ [x : y : z] → [xz − yz : x2 : y2 : z2] (Q6)✿ [x : y : z] → [x2 : xz − y2 : yz : z2] (Q7)✿ [x : y : z] → [y2 − z2 : xy : xz : yz] (Q8)✿ [x : y : z] → [xy : xz : y2 : z2] (Q9)✿ [x : y : z] → [xy : xz − y2 : yz : z2] (Q10)✿ [x : y : z] → [x2 : xy : y2 : z2] . . .

❚❤❡ ❝❧❛ss✐✜❝❛t✐♦♥

(C1)✿ [x : y : z] → [z(x2 + y2) : y(x2 + z2) : x(y2 + z2) : xyz] (C2)✿ [x : y : z] → [z(x2 − y2) : y(x2 + z2) : x(y2 − z2) : xyz] (C3)✿ [x : y : z] → [x2z : z(x2+y2) : x(x2+y2−z2) : y(x2+y2+z2)] (C4)✿ [x : y : z] → [x2y : x(x2 − y2) : z(x2 + y2) : yz2] (C5)✿ [x : y : z] → [x2(x + y) : y2(x + y) : z2(x − y) : xyz] (C6)✿ [x : y : z] → [x3 : xy2 : 2xyz − y3 : z(xz − y2)]✳

P❧❛♥❛r✐③❛t✐♦♥s (C1) ❛♥❞ (C2)

P❧❛♥❛r✐③❛t✐♦♥s (C3) ❛♥❞ (C4)

P❧❛♥❛r✐③❛t✐♦♥s (C5) ❛♥❞ (C6)