Quadratic Bézier curve pointsA quadratic Bézier curve is the path traced by the function B(t), given points P 0 , P 1 , and P 2 . C ( t ) = ∑ i = 0 2 ( 2 i ) t i ( 1 − t ) 2 − i P i C ( t ) = ( P 0 − 2 P 1 + P 2 ) t 2 + ( − 2 P 0 + 2 P 1 ) t + P 0 t ∈ [ 0 , 1 ]What exactly is P 0 or P 1 or P 2 concerning this equation?Yes they are points. But in my understanding, a point is a pair of numbers (in 2D-space).Let = ( 1 , 1 ), P 1 = ( 1 , 7 ) and P 2 = ( 7 , 1 ).What values do you use (and where)?How do you calculate the Bezier Curve for these points?

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Quadratic Bézier curve pointsA quadratic Bézier curve is the path traced by the function B(t), given points       P    0  ,       P    1  , and       P    2  .  C  (  t  )  =      ∑          i      =      0              2                          (                    2      i                      )                  t    i    (  1  −  t      )          2      −      i            P    i    C  (  t  )  =  (      P    0    −  2      P    1    +      P    2    )      t    2    +  (  −  2      P    0    +  2      P    1    )  t  +      P    0      t  ∈  [  0  ,  1  ]What exactly is       P    0   or       P    1   or       P    2   concerning this equation?Yes they are points. But in my understanding, a point is a pair of numbers (in 2D-space).Let   =  (  1  ,  1  ),       P    1    =  (  1  ,  7  ) and       P    2    =  (  7  ,  1  ).What values do you use (and where)?How do you calculate the Bezier Curve for these points?

lavarcar2d2 · Accepted Answer

Step 1You can use standard operations on points in natural ways.Adding (or subtracting) two points:       P    1    +      P    2   =   (      x    1    +      x    2    ,         y    1    +      y    2    )Multiplying (or dividing) a point by a number:       P    1    k  =  (      x    1    k  ,         y    1    k  ).The Bézier curve uses only the standard numerical operators and these two additional abilities.Step 2an example, using your data.  C  (  t  )  =  (      P    0    −  2      P    1    +      P    2    )      t    2    +  (  −  2      P    0    +  2      P    1    )  t  +      P    0      t  ∈  [  0  ,  1  ]  C  (  t  )  =  (  (  1  ,  1  )  −  2  (  1  ,  7  )  +  (  7  ,  1  )  )      t    2    +  (  −  2  (  1  ,  1  )  +  2  (  1  ,  7  )  )  t  +  (  1  ,  1  )Applying multiplication:  C  (  t  )  =  (  (  1  ,  1  )  −  (  2  ,  14  )  +  (  7  ,  1  )  )      t    2    +  (  (  −  2  ,  −  2  )  +  (  2  ,  14  )  )  t  +  (  1  ,  1  )And addition:  C  (  t  )  =  (  6  ,  −  12  )      t    2    +  (  0  ,  12  )  t  +  (  1  ,  1  )then we can handle it by components by multiplying and adding some more:  C  (  t  )  =  (  6      t    2    +  1  ,  −  12      t    2    +  12  t  +  1  )Usually you won&#039;t see it shown this way, though: The operations described above are considered already natural and straightforward.

pighead73283r · Accepted Answer

Step 1Generally, the formula should read (for your special case, the quadratic,   m  =  2  )  C  (  t  )  =      ∑          i      =      0        m        (                            m                                      i                      )        P    i    (  1  −  t      )    i        t          m      −      i      In a 2-dimensional environment, with control points   (      x    i    ∣      y    i    )    i  =  {  0  ,  1  ,  2  }  ,,  m  =  2.You would need TWO instances of the formula: one for each dimension. The first instance would utilize the x parts of your given control points in place of the Ps. The second instance would utilize the y parts. For each value of t, the two values yielded by these TWO instances would constitute the two coördinates for each point of your desired Bézier curve. Clear, I hope?Step 2For your example we need two coördinates for each of your given points. I have assigned each coördinate a name, as follows:                    point                    x                    y                                      P          0                                      P                      0            ,            x                                                P                      0            ,            y                                                        P          1                                      P                      1            ,            x                                                P                      1            ,            y                                                        P          2                                      P                      2            ,            x                                                P                      2            ,            y                              For each value of t, the corresponding point of your desired curve is   (      x    t    ∣      y    t    )  ,   where      x    t    =      P          0      ,      x        (  1  −  t      )    2    +  2      P          1      ,      x        (  1  −  t  )  t  +      P          2      ,      x            t    2        y    t    =      P          0      ,      y        (  1  −  t      )    2    +  2      P          1      ,      y        (  1  −  t  )  t  +      P          2      ,      y            t    2

A quadratic Bézier curve is the path traced by the function B(t), given points P_0, P_1, and P_2.

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