diff --git a/geo/src/algorithm/simplify.rs b/geo/src/algorithm/simplify.rs
index a829134432..aa63cc0a1f 100644
--- a/geo/src/algorithm/simplify.rs
+++ b/geo/src/algorithm/simplify.rs
@@ -19,6 +19,10 @@ fn rdp<T>(points: &[Point<T>], epsilon: &T) -> Vec<Point<T>>
 where
     T: Float,
 {
+    // Epsilon must be greater than zero for any meaningful simplification to happen
+    if *epsilon <= T::zero() {
+        points.to_vec();
+    }
     compute_rdp(
         &points
             .into_iter()
diff --git a/geo/src/algorithm/simplifyvw.rs b/geo/src/algorithm/simplifyvw.rs
index eaf3aa3908..23379ac9ea 100644
--- a/geo/src/algorithm/simplifyvw.rs
+++ b/geo/src/algorithm/simplifyvw.rs
@@ -191,6 +191,10 @@ fn visvalingam<T>(orig: &LineString<T>, epsilon: &T) -> Vec<Coordinate<T>>
 where
     T: Float,
 {
+    // Epsilon must be greater than zero for any meaningful simplification to happen
+    if *epsilon <= T::zero() {
+        return orig.0.to_vec()
+    }
     let subset = visvalingam_indices(orig, epsilon);
     // filter orig using the indices
     // using get would be more robust here, but the input subset is guaranteed to be valid in this case
@@ -251,7 +255,7 @@ fn visvalingam_preserve<T>(
 where
     T: Float + RTreeNum,
 {
-    if orig.0.len() < 3 {
+    if orig.0.len() < 3 || *epsilon <= T::zero() {
         return orig.0.to_vec();
     }
     let max = orig.0.len();