diff --git a/docs/changelog.rst b/docs/changelog.rst
index 1952ec320..4c5771fdd 100644
--- a/docs/changelog.rst
+++ b/docs/changelog.rst
@@ -59,6 +59,8 @@ To update |kitty|, :doc:`follow the instructions <binary>`.
 - Add an option :option:`kitty @ get-text --add-wrap-markers` to add line wrap
   markers to the output (:pull:`3633`)
 
+- Improve rendering of curly underlines on HiDPI screens (:pull:`3637`)
+
 - ssh kitten: Mimic behavior of ssh command line client more closely by
   executing any command specified on the command line via the users' shell
   just as ssh does (:iss:`3638`)
diff --git a/kitty/fonts/render.py b/kitty/fonts/render.py
index b48cc8bdf..8be9947f5 100644
--- a/kitty/fonts/render.py
+++ b/kitty/fonts/render.py
@@ -236,7 +236,15 @@ def add_dline(buf: ctypes.Array, cell_width: int, position: int, thickness: int,
 def add_curl(buf: ctypes.Array, cell_width: int, position: int, thickness: int, cell_height: int) -> None:
     max_x, max_y = cell_width - 1, cell_height - 1
     xfactor = 2.0 * pi / max_x
-    half_height = max(thickness // 2, 1)
+    thickness = max(1, thickness)
+    if thickness < 3:
+        half_height = thickness
+        thickness -= 1
+    elif thickness == 3:
+        half_height = thickness = 2
+    else:
+        half_height = thickness // 2
+        thickness -= 2
 
     def add_intensity(x: int, y: int, val: int) -> None:
         y += position
@@ -244,20 +252,22 @@ def add_curl(buf: ctypes.Array, cell_width: int, position: int, thickness: int,
         idx = cell_width * y + x
         buf[idx] = min(255, buf[idx] + val)
 
-    # Ensure all space at bottom of cell is used
-    if position + half_height < max_y:
-        position += max_y - (position + half_height)
+    # Ensure curve doesn't exceed cell boundary at the bottom
+    position += half_height * 2
     if position + half_height > max_y:
-        position -= position + half_height - max_y
+        position = max_y - half_height
 
     # Use the Wu antialias algorithm to draw the curve
     # cosine waves always have slope <= 1 so are never steep
     for x in range(cell_width):
         y = half_height * cos(x * xfactor)
-        y1, y2 = floor(y), ceil(y)
-        i1 = int(255 * abs(y - y1))
-        add_intensity(x, y1, 255 - i1)
-        add_intensity(x, y2, i1)
+        y1, y2 = floor(y - thickness), ceil(y)
+        i1 = int(255 * abs(y - floor(y)))
+        add_intensity(x, y1, 255 - i1)  # upper bound
+        add_intensity(x, y2, i1)  # lower bound
+        # fill between upper and lower bound
+        for t in range(1, thickness + 1):
+            add_intensity(x, y1 + t, 255)
 
 
 def render_special(