diff --git a/libs/mathlib.h b/libs/mathlib.h
index ae98071e..aafe568d 100644
--- a/libs/mathlib.h
+++ b/libs/mathlib.h
@@ -38,12 +38,12 @@ typedef vec_t vec3_t[3];
 typedef vec_t vec5_t[5];
 typedef vec_t vec4_t[4];
 
-// Smallest positive value such that 1.0 + VEC_SMALLEST_EPSILON != 1.0
-// In the case of 32 bits (which is the case), it's 0.00000011921.
+// Smallest positive value such that 1.0 + VEC_SMALLEST_EPSILON_AROUND_ONE != 1.0.
+// In the case of 32 bits (which is likely the case), it's 0.00000011921.
 // Don't forget that your epsilons should depend on the possible range of values,
-// because for example 1000.0 + VEC_SMALLEST_EPSILON will almost certainly be
-// equal to 1000.0.
-#define VEC_SMALLEST_EPSILON FLT_EPSILON
+// because for example adding VEC_SMALLEST_EPSILON_AROUND_ONE to 1024.0 will almost
+// certainly not change its value.
+#define VEC_SMALLEST_EPSILON_AROUND_ONE FLT_EPSILON
 
 #define	SIDE_FRONT		0
 #define	SIDE_ON			2
@@ -316,12 +316,12 @@ vec_t ray_intersect_triangle(const ray_t *ray, qboolean bCullBack, const vec3_t
 typedef double vec_accu_t;
 typedef vec_accu_t vec3_accu_t[3];
 
-// Smallest positive value such that 1.0 + VEC_ACCU_SMALLEST_EPSILON != 1.0
-// In the case of 64 bits (which is the case), it's 0.00000000000000022204.
+// Smallest positive value such that 1.0 + VEC_ACCU_SMALLEST_EPSILON_AROUND_ONE != 1.0.
+// In the case of 64 bits (which is likely the case), it's 0.00000000000000022204.
 // Don't forget that your epsilons should depend on the possible range of values,
-// because for example 1000.0 + VEC_ACCU_SMALLEST_EPSILON will almost certainly
-// be equal to 1000.0.
-#define VEC_ACCU_SMALLEST_EPSILON DBL_EPSILON
+// because for example adding VEC_ACCU_SMALLEST_EPSILON_AROUND_ONE to 1024.0 will almost
+// certainly not change its value.
+#define VEC_ACCU_SMALLEST_EPSILON_AROUND_ONE DBL_EPSILON
 
 // TODO: I have a feeling it may be safer to break these function out into actual functions
 // in order to avoid accidental loss of precision.  For example, say you call
diff --git a/tools/quake3/common/polylib.c b/tools/quake3/common/polylib.c
index 93f15241..79eb3fb1 100644
--- a/tools/quake3/common/polylib.c
+++ b/tools/quake3/common/polylib.c
@@ -632,7 +632,18 @@ void ChopWindingInPlaceAccu(winding_accu_t **inout, vec3_t normal, vec_t dist, v
 	// want the resolution of vec_accu_t to have a large resolution around the epsilon.
 	// Some of that leftover resolution even goes away after we scale to points far away.
 
-	static const vec_accu_t smallestEpsilonAllowed = ((vec_accu_t) VEC_SMALLEST_EPSILON) * 0.5;
+	// Here is a further discussion regarding the choice of smallestEpsilonAllowed.
+	// In the 32 float world (we can assume vec_t is that), the "epsilon around 1.0" is
+	// 0.00000011921.  In the 64 bit float world (we can assume vec_accu_t is that), the
+	// "epsilon around 1.0" is 0.00000000000000022204.  (By the way these two epsilons
+	// are defined as VEC_SMALLEST_EPSILON_AROUND_ONE VEC_ACCU_SMALLEST_EPSILON_AROUND_ONE
+	// respectively.)  If you divide the first by the second, you get approximately
+	// 536,885,246.  Dividing that number by 200,000 (a typical base winding coordinate)
+	// gives 2684.  So in other words, if our smallestEpsilonAllowed was chosen as exactly
+	// VEC_SMALLEST_EPSILON_AROUND_ONE, you would be guaranteed at least 2000 "ticks" in
+	// 64-bit land inside of the epsilon for all numbers we're dealing with.
+
+	static const vec_accu_t smallestEpsilonAllowed = ((vec_accu_t) VEC_SMALLEST_EPSILON_AROUND_ONE) * 0.5;
 	if (crudeEpsilon < smallestEpsilonAllowed)	fineEpsilon = smallestEpsilonAllowed;
 	else						fineEpsilon = (vec_accu_t) crudeEpsilon;