3.10 Discontinuities

There are several functions which “clamp” their inputs in various ways, i.e. potentially lose some information in their inputs and introduce discontinuities. These are documented in Tables 3.21, 3.22, and 3.23. The first set of functions are sharp discontinuities, the second supports smoothed discontinuities, as required in shaders for antialiasing. The smoothed version of a function typically has smooth in front of the name and an extra parameter for setting the width of the transition region.


Operation	r a,b,c,...	Description


abs	N N	Componentwise absolute value $ri <-- \|ai\|$

ceil	N N	Componentwise ceiling $r <-- \|~ a ~\| i i$

/ clamp	N N,N,N	Componentwise clamping of a_i between b_i and c_i

/ clamp	N N,1,1	Componentwise clamping of a_i between b_i and c_i

floor	N N	Componentwise floor $ri <-- \|_ ai _\|$

/ frac	N N	Componentwise fractional part $ri <-- ai- \|_ \| ai\| _\|$

/ mod	N N,N	Componentwise floating-point modulus $ri <-- ai/bi- \|_ ai/bi _\|$

/ round	N N	Componentwise nearest integer $ri <-- \|_ ai + 1/2 _\|$

/ sign	N N	Componentwise sign $ri <-- ai/ \|ai\|$

Table 3.21:

Clamping functions. These operations all potentially lose information and introduce discontinuities.


Operation	r a,b,c,...	Description


max	N N,N	Componentwise maximum $ri <-- max(ai,bi)$

/ max	1 N	Tuple maximum $r <-- max(a ,a ,...,a ) 0 1 N -1$

min	N N,N	Componentwise minimum $ri <-- min(ai,bi)$

/ min	1 N	Tuple minimum $r <-- min(a0,a1,...,aN- 1)$

pos	N N	Componentwise maximum with 0

/ sat	N N	Componentwise minimum with 1

Table 3.22:

Minimum and maximum functions.


Operation	r a,b,c,...	Description


/ smoothstep	N N,N,N	Linear smooth step of a_i centered around b_i with width c_i

/ smoothpulse	N N,N,N,N	Linear smooth pulse of a_i between b_i and c_i with width d_i

Table 3.23:

Smooth clamping functions.

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