Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

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Essential Geometry

Transcript of Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Page 1: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Essential Geometry

Page 2: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Essential Geometry

Page 3: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Essential Geometry

Page 4: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Essential Geometry

Page 5: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Essential Geometry

Page 6: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Equations

• Implicit– F(x,y) = 0

• Parametric– P(t) = x– Q(t) = y– 0 <= t <= 1

Page 7: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric Operations

• Scan conversion– Convert geometry into pixels– Let the graphics system do it

• Bounding box – Efficiency

• Distance from a point– Selection

• Nearest Point– Snapping

• Inside/Outside– Selection

Page 8: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Points and Vectors

dX

dYX,Y

Page 9: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric Operations

Page 10: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric OperationsTop

Left Right

Bottom

Page 11: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric Operations

X1,Y1

X1,Y1

ImplicitParametric

Page 12: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric Operations

Cx,CyR

ImplicitParametric

Page 13: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric OperationsTop

Left Right

Bottom

Page 14: Essential Geometry. Equations Implicit – F(x,y) = 0 Parametric – P(t) = x – Q(t) = y – 0

Geometric Operations

B

Cx,CyA

ImplicitParametric


x y z { | } ~ ~ } { | | } { { | | } | } | { z y { y ... · 2018-01-23 · " / 0 , ! % ! ' ( # + % * " 0 , ! %

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Revision sheet Grade 9 Mathematics...If (x) (y) 0, then If (x 3) (x 2) 0, then x 0 or y 0 x 3 0 or x 2 0 You can use the Zero Product Property to solve any quadratic equation written

Revision sheet Grade 9 Mathematics...If (x) (y) 0, then If (x 3) (x 2) 0, then x 0 or y 0 x 3 0 or x 2 0 You can use the Zero Product Property to solve any quadratic equation written

arXiv:2002.06277v1 [cs.LG] 14 Feb 2020 · Algorithm1LangevinDescent-Ascent. 1: Input: IIDsamplesx1 0;:::;x n 0from x; 2P(X),IIDsamplesy1 0;:::;y n 0 2Yfrom y; 2P(Y) 2: fort= 0;:::;Tdo

arXiv:2002.06277v1 [cs.LG] 14 Feb 2020 · Algorithm1LangevinDescent-Ascent. 1: Input: IIDsamplesx1 0;:::;x n 0from x; 2P(X),IIDsamplesy1 0;:::;y n 0 2Yfrom y; 2P(Y) 2: fort= 0;:::;Tdo

実施設計db.pref.tottori.jp/HacchuKoukai.nsf...Y = 1 0 0 Y = 1 0 0 0 Y = 1 1 0 Y = 1 1 0 0 Y = 1 2 0 0 Y = 1 2 0 0 Y = 1 3 0 0 Y = 1 4 0 0 X = - 4 6 00 0 X =-X-4 5 90 0 X =-X =-4

実施設計db.pref.tottori.jp/HacchuKoukai.nsf...Y = 1 0 0 Y = 1 0 0 0 Y = 1 1 0 Y = 1 1 0 0 Y = 1 2 0 0 Y = 1 2 0 0 Y = 1 3 0 0 Y = 1 4 0 0 X = - 4 6 00 0 X =-X-4 5 90 0 X =-X =-4

X= {x 0, x 1,….,x J-1 } Y= {y 0, y 1, ….,y K-1 } Channel Finite set of input (X= {x 0, x 1,….,x J-1 }), and output (Y= {y 0, y 1,….,y K-1 }) alphabet.

X= {x 0, x 1,….,x J-1 } Y= {y 0, y 1, ….,y K-1 } Channel Finite set of input (X= {x 0, x 1,….,x J-1 }), and output (Y= {y 0, y 1,….,y K-1 }) alphabet.

X + 0 + Y 0MHZ - support.jbl.com

X + 0 + Y 0MHZ - support.jbl.com

~ 0 [u(x,y)/U e ] (1 – u(x,y)/U e )dy

~ 0 [u(x,y)/U e ] (1 – u(x,y)/U e )dy

Mathematics in Education and Industry · 2018-04-04 · 18 6 =0 when 3 0 d 30 30 d 18 12 d d When 3 0 Maximum d When 3 27 y xx x x x xx x , y x x y x x x,y =− −= ⇒−=⇒= =−

Mathematics in Education and Industry · 2018-04-04 · 18 6 =0 when 3 0 d 30 30 d 18 12 d d When 3 0 Maximum d When 3 27 y xx x x x xx x , y x x y x x x,y =− −= ⇒−=⇒= =−

6.641 Electromagnetic Fields, Forces, and Motion · Ky1 x +∞ =− 0 tan −1 2π y y x=−∞ ⎧ K ⎪− 0 y > 0 ⎪ 2 =⎨ ⎪+ K 0 y < 0 ⎪⎩ 2 Check boundary condition

6.641 Electromagnetic Fields, Forces, and Motion · Ky1 x +∞ =− 0 tan −1 2π y y x=−∞ ⎧ K ⎪− 0 y > 0 ⎪ 2 =⎨ ⎪+ K 0 y < 0 ⎪⎩ 2 Check boundary condition

Reasoning - Massachusetts Institute of Technologypeople.csail.mit.edu/cpacheco/publications/reasoning-tla.pdf · Init = (x 0) ^ y M 1 = (x 0 + 1) ^ y) M 2 = (y 0 + 1) ^ x) M = 1 _

Reasoning - Massachusetts Institute of Technologypeople.csail.mit.edu/cpacheco/publications/reasoning-tla.pdf · Init = (x 0) ^ y M 1 = (x 0 + 1) ^ y) M 2 = (y 0 + 1) ^ x) M = 1 _

BigHorn- Y-- B YIJ--X 0

BigHorn- Y-- B YIJ--X 0

S.4 Mathematics x + y –7 = 0 2x – 3y +6=0 x y 0 (3, 4) Put (3,4) into x +y –7 =0 LHS = 3+4 – 7 Put (3,4) into 2x –3y +6 =0 LHS = (2)3 – 3(4) + 6 = 0.

S.4 Mathematics x + y –7 = 0 2x – 3y +6=0 x y 0 (3, 4) Put (3,4) into x +y –7 =0 LHS = 3+4 – 7 Put (3,4) into 2x –3y +6 =0 LHS = (2)3 – 3(4) + 6 = 0.

Optimal Design of Micro-Seismic Monitoring Array and ...0 = (x 0 x i)2 +(y 0 y i)2 +(z 0 z i)2 v(x 0,y 0,z 0) (1) Where x 0, y 0 and z 0 are the hypocenter coordinates, t 0 is the

Optimal Design of Micro-Seismic Monitoring Array and ...0 = (x 0 x i)2 +(y 0 y i)2 +(z 0 z i)2 v(x 0,y 0,z 0) (1) Where x 0, y 0 and z 0 are the hypocenter coordinates, t 0 is the

Graphing Inverse Variation Functions - Weebly€¦ · Notes p. 2 2 2 4. ( 5) y x 5. y x 2 6. ( 1) y x VA:_____ HA:_____ VA:_____ HA:_____ VA:_____ HA:_____ x 5 y 0 y 0 x 0 x 1 y 0.

Graphing Inverse Variation Functions - Weebly€¦ · Notes p. 2 2 2 4. ( 5) y x 5. y x 2 6. ( 1) y x VA:_____ HA:_____ VA:_____ HA:_____ VA:_____ HA:_____ x 5 y 0 y 0 x 0 x 1 y 0.

Multiplicadores de Lagrange · Multiplicadores de Lagrange (2 2 ) 0 (2 2 ) 0 (2 2 ) 0 yz y z xz x z xy y x g x y z xy yz xz( , , ) 2 2 2 64 0 ... Multiplicadores de Lagrange f x y

Multiplicadores de Lagrange · Multiplicadores de Lagrange (2 2 ) 0 (2 2 ) 0 (2 2 ) 0 yz y z xz x z xy y x g x y z xy yz xz( , , ) 2 2 2 64 0 ... Multiplicadores de Lagrange f x y

Unit 5 x -5 0 510 0 y 20 x -5 0 510 0 y 20 x -5 0 510 0 y 20 x -5 0 510 0 y 20 Check this 10.00 = 10(1.1) 0 = 10(1+.1) 0 11.00 = 10(1.1) 1 = 10(1+.1) 1.

Unit 5 x -5 0 510 0 y 20 x -5 0 510 0 y 20 x -5 0 510 0 y 20 x -5 0 510 0 y 20 Check this 10.00 = 10(1.1) 0 = 10(1+.1) 0 11.00 = 10(1.1) 1 = 10(1+.1) 1.

Series 2/3 - RWTH Aachen University · 2017. 10. 4. · x 12 x := 0 y := 0 fwaitg x 2 frecg x 5 ffwdg x = 0 fackg y 13 fdistg x 7 reset(x) x 1 reset(x) x 3 reset(x) reset(y) y 10

Series 2/3 - RWTH Aachen University · 2017. 10. 4. · x 12 x := 0 y := 0 fwaitg x 2 frecg x 5 ffwdg x = 0 fackg y 13 fdistg x 7 reset(x) x 1 reset(x) x 3 reset(x) reset(y) y 10

12.4 8-Point Algorithm16385/s17/Slides/12.4_8Point_Algorithm.pdf · 2017. 4. 5. · 8 f 9 3 5 2 4 x m y m 1 3 5 =0 2 6 4 x 1x0 1 x 1y 0 1 x 1 y 1x 0 1 y 1y 0 1 y 1 x 0 1 y 0 ... Construct

12.4 8-Point Algorithm16385/s17/Slides/12.4_8Point_Algorithm.pdf · 2017. 4. 5. · 8 f 9 3 5 2 4 x m y m 1 3 5 =0 2 6 4 x 1x0 1 x 1y 0 1 x 1 y 1x 0 1 y 1y 0 1 y 1 x 0 1 y 0 ... Construct

Lecture2 BerndFinkbeiner PeterFaymonville MichaelGerke › teaching › ... · REVIEW:Modelingsequentialcircuits XOR OR f y g NOT x f r g x; r y x = 0 r 0 x = 0 r 1 x 1 r 0 x 1 r

Lecture2 BerndFinkbeiner PeterFaymonville MichaelGerke › teaching › ... · REVIEW:Modelingsequentialcircuits XOR OR f y g NOT x f r g x; r y x = 0 r 0 x = 0 r 1 x 1 r 0 x 1 r