forked from minhngoc25a/freetype2
* src/sdf/ftsdf.c: Use ASCII single quote (').
This commit is contained in:
Anuj Verma
anujverma
parent
ddf631729e
commit
4fcc1653ca
|
|
@ -1,3 +1,8 @@
|
|||
2020-07-03 Anuj Verma <anujv@iitbhilai.ac.in>
|
||||
|
||||
* src/sdf/ftsdf.c: Use ASCII single quote (') instead
|
||||
of back tick (`) for derivatives. Looks cleaner.
|
||||
|
||||
2020-07-03 Anuj Verma <anujv@iitbhilai.ac.in>
|
||||
|
||||
[sdf] Added function to find shortest distance from a
|
||||
|
|
|
|||
|
|
@ -1214,14 +1214,14 @@
|
|||
/* B( t ) = t^2( A ) + 2t( B ) + p0 */
|
||||
/* */
|
||||
/* => the derivative of the above equation is written as */
|
||||
/* B`( t ) = 2( tA + B ) */
|
||||
/* B'( t ) = 2( tA + B ) */
|
||||
/* */
|
||||
/* => now to find the shortest distance from p to B( t ), we */
|
||||
/* find the point on the curve at which the shortest */
|
||||
/* distance vector ( i.e. B( t ) - p ) and the direction */
|
||||
/* ( i.e. B`( t )) makes 90 degrees. in other words we make */
|
||||
/* ( i.e. B'( t )) makes 90 degrees. in other words we make */
|
||||
/* the dot product zero. */
|
||||
/* ( B( t ) - p ).( B`( t ) ) = 0 */
|
||||
/* ( B( t ) - p ).( B'( t ) ) = 0 */
|
||||
/* ( t^2( A ) + 2t( B ) + p0 - p ).( 2( tA + B ) ) = 0 */
|
||||
/* */
|
||||
/* after simplifying we get a cubic equation as */
|
||||
|
|
@ -1372,7 +1372,7 @@
|
|||
}
|
||||
}
|
||||
|
||||
/* B`( t ) = 2( tA + B ) */
|
||||
/* B'( t ) = 2( tA + B ) */
|
||||
direction.x = 2 * FT_MulFix( aA.x, min_factor ) + 2 * bB.x;
|
||||
direction.y = 2 * FT_MulFix( aA.y, min_factor ) + 2 * bB.y;
|
||||
|
||||
|
|
@ -1439,10 +1439,10 @@
|
|||
/* B( t ) = t^2( A ) + t( B ) + p0 */
|
||||
/* */
|
||||
/* => the derivative of the above equation is written as */
|
||||
/* B`( t ) = 2t( A ) + B */
|
||||
/* B'( t ) = 2t( A ) + B */
|
||||
/* */
|
||||
/* => further derivative of the above equation is written as */
|
||||
/* B``( t ) = 2A */
|
||||
/* B''( t ) = 2A */
|
||||
/* */
|
||||
/* => the equation of distance from point `p' to the curve */
|
||||
/* P( t ) can be written as */
|
||||
|
|
@ -1452,7 +1452,7 @@
|
|||
/* */
|
||||
/* => finally the equation of angle between curve B( t ) and */
|
||||
/* point to curve distance P( t ) can be written as */
|
||||
/* Q( t ) = P( t ).B`( t ) */
|
||||
/* Q( t ) = P( t ).B'( t ) */
|
||||
/* */
|
||||
/* => now our task is to find a value of t such that the above */
|
||||
/* equation Q( t ) becomes zero. in other words the point */
|
||||
|
|
@ -1462,19 +1462,19 @@
|
|||
/* => we first assume a arbitary value of the factor `t' and */
|
||||
/* then we improve it using Newton's equation such as */
|
||||
/* */
|
||||
/* t -= Q( t ) / Q`( t ) */
|
||||
/* t -= Q( t ) / Q'( t ) */
|
||||
/* putting value of Q( t ) from the above equation gives */
|
||||
/* */
|
||||
/* t -= P( t ).B`( t ) / derivative( P( t ).B`( t ) ) */
|
||||
/* t -= P( t ).B`( t ) / */
|
||||
/* ( P`( t )B`( t ) + P( t ).B``( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / derivative( P( t ).B'( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / */
|
||||
/* ( P'( t )B'( t ) + P( t ).B''( t ) ) */
|
||||
/* */
|
||||
/* P`( t ) is noting but B`( t ) because the constant are */
|
||||
/* P'( t ) is noting but B'( t ) because the constant are */
|
||||
/* gone due to derivative */
|
||||
/* */
|
||||
/* => finally we get the equation to improve the factor as */
|
||||
/* t -= P( t ).B`( t ) / */
|
||||
/* ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / */
|
||||
/* ( B'( t ).B'( t ) + P( t ).B''( t ) ) */
|
||||
/* */
|
||||
/* [note]: B and B( t ) are different in the above equations */
|
||||
|
||||
|
|
@ -1567,24 +1567,24 @@
|
|||
}
|
||||
|
||||
/* This the actual Newton's approximation. */
|
||||
/* t -= P( t ).B`( t ) / */
|
||||
/* ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / */
|
||||
/* ( B'( t ).B'( t ) + P( t ).B''( t ) ) */
|
||||
|
||||
/* B`( t ) = 2tA + B */
|
||||
/* B'( t ) = 2tA + B */
|
||||
d1.x = FT_MulFix( aA.x, 2 * factor ) + bB.x;
|
||||
d1.y = FT_MulFix( aA.y, 2 * factor ) + bB.y;
|
||||
|
||||
/* B``( t ) = 2A */
|
||||
/* B''( t ) = 2A */
|
||||
d2.x = 2 * aA.x;
|
||||
d2.y = 2 * aA.y;
|
||||
|
||||
dist_vector.x /= 1024;
|
||||
dist_vector.y /= 1024;
|
||||
|
||||
/* temp1 = P( t ).B`( t ) */
|
||||
/* temp1 = P( t ).B'( t ) */
|
||||
temp1 = VEC_26D6_DOT( dist_vector, d1 );
|
||||
|
||||
/* temp2 = ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
|
||||
/* temp2 = ( B'( t ).B'( t ) + P( t ).B''( t ) ) */
|
||||
temp2 = VEC_26D6_DOT( d1, d1 ) +
|
||||
VEC_26D6_DOT( dist_vector, d2 );
|
||||
|
||||
|
|
@ -1595,7 +1595,7 @@
|
|||
}
|
||||
}
|
||||
|
||||
/* B`( t ) = 2tA + B */
|
||||
/* B'( t ) = 2tA + B */
|
||||
direction.x = 2 * FT_MulFix( aA.x, min_factor ) + bB.x;
|
||||
direction.y = 2 * FT_MulFix( aA.y, min_factor ) + bB.y;
|
||||
|
||||
|
|
@ -1670,10 +1670,10 @@
|
|||
/* B( t ) = t^3( A ) + t^2( B ) + tC + p0 */
|
||||
/* */
|
||||
/* => the derivative of the above equation is written as */
|
||||
/* B`( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
/* B'( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
/* */
|
||||
/* => further derivative of the above equation is written as */
|
||||
/* B``( t ) = 6t( A ) + 2B */
|
||||
/* B''( t ) = 6t( A ) + 2B */
|
||||
/* */
|
||||
/* => the equation of distance from point `p' to the curve */
|
||||
/* P( t ) can be written as */
|
||||
|
|
@ -1683,7 +1683,7 @@
|
|||
/* */
|
||||
/* => finally the equation of angle between curve B( t ) and */
|
||||
/* point to curve distance P( t ) can be written as */
|
||||
/* Q( t ) = P( t ).B`( t ) */
|
||||
/* Q( t ) = P( t ).B'( t ) */
|
||||
/* */
|
||||
/* => now our task is to find a value of t such that the above */
|
||||
/* equation Q( t ) becomes zero. in other words the point */
|
||||
|
|
@ -1693,19 +1693,19 @@
|
|||
/* => we first assume a arbitary value of the factor `t' and */
|
||||
/* then we improve it using Newton's equation such as */
|
||||
/* */
|
||||
/* t -= Q( t ) / Q`( t ) */
|
||||
/* t -= Q( t ) / Q'( t ) */
|
||||
/* putting value of Q( t ) from the above equation gives */
|
||||
/* */
|
||||
/* t -= P( t ).B`( t ) / derivative( P( t ).B`( t ) ) */
|
||||
/* t -= P( t ).B`( t ) / */
|
||||
/* ( P`( t )B`( t ) + P( t ).B``( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / derivative( P( t ).B'( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / */
|
||||
/* ( P'( t )B'( t ) + P( t ).B''( t ) ) */
|
||||
/* */
|
||||
/* P`( t ) is noting but B`( t ) because the constant are */
|
||||
/* P'( t ) is noting but B'( t ) because the constant are */
|
||||
/* gone due to derivative */
|
||||
/* */
|
||||
/* => finally we get the equation to improve the factor as */
|
||||
/* t -= P( t ).B`( t ) / */
|
||||
/* ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / */
|
||||
/* ( B'( t ).B'( t ) + P( t ).B''( t ) ) */
|
||||
/* */
|
||||
/* [note]: B and B( t ) are different in the above equations */
|
||||
|
||||
|
|
@ -1809,26 +1809,26 @@
|
|||
}
|
||||
|
||||
/* This the actual Newton's approximation. */
|
||||
/* t -= P( t ).B`( t ) / */
|
||||
/* ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
|
||||
/* t -= P( t ).B'( t ) / */
|
||||
/* ( B'( t ).B'( t ) + P( t ).B''( t ) ) */
|
||||
|
||||
/* B`( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
/* B'( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
d1.x = FT_MulFix( aA.x, 3 * factor2 ) +
|
||||
FT_MulFix( bB.x, 2 * factor ) + cC.x;
|
||||
d1.y = FT_MulFix( aA.y, 3 * factor2 ) +
|
||||
FT_MulFix( bB.y, 2 * factor ) + cC.y;
|
||||
|
||||
/* B``( t ) = 6t( A ) + 2B */
|
||||
/* B''( t ) = 6t( A ) + 2B */
|
||||
d2.x = FT_MulFix( aA.x, 6 * factor ) + 2 * bB.x;
|
||||
d2.y = FT_MulFix( aA.y, 6 * factor ) + 2 * bB.y;
|
||||
|
||||
dist_vector.x /= 1024;
|
||||
dist_vector.y /= 1024;
|
||||
|
||||
/* temp1 = P( t ).B`( t ) */
|
||||
/* temp1 = P( t ).B'( t ) */
|
||||
temp1 = VEC_26D6_DOT( dist_vector, d1 );
|
||||
|
||||
/* temp2 = ( B`( t ).B`( t ) + P( t ).B``( t ) ) */
|
||||
/* temp2 = ( B'( t ).B'( t ) + P( t ).B''( t ) ) */
|
||||
temp2 = VEC_26D6_DOT( d1, d1 ) +
|
||||
VEC_26D6_DOT( dist_vector, d2 );
|
||||
|
||||
|
|
@ -1839,7 +1839,7 @@
|
|||
}
|
||||
}
|
||||
|
||||
/* B`( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
/* B'( t ) = 3t^2( A ) + 2t( B ) + C */
|
||||
direction.x = FT_MulFix( aA.x, 3 * min_factor_sq ) +
|
||||
FT_MulFix( bB.x, 2 * min_factor ) + cC.x;
|
||||
direction.y = FT_MulFix( aA.y, 3 * min_factor_sq ) +
|
||||
|
|
|
|||
Loading…
Reference in New Issue