diff --git a/engine-proto/gram-test/Engine.jl b/engine-proto/gram-test/Engine.jl
index 2662a17..40b77b0 100644
--- a/engine-proto/gram-test/Engine.jl
+++ b/engine-proto/gram-test/Engine.jl
@@ -41,7 +41,7 @@ rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), she
 
 point(pos) = [pos; 0.5; 0.5 * dot(pos, pos)]
 
-plane(normal, offset) = [normal; 0; offset]
+plane(normal, offset) = [-normal; 0; -offset]
 
 function sphere(center, radius)
   dist_sq = dot(center, center)
diff --git a/engine-proto/gram-test/circles-in-triangle.jl b/engine-proto/gram-test/circles-in-triangle.jl
index fc5e13d..ca49574 100644
--- a/engine-proto/gram-test/circles-in-triangle.jl
+++ b/engine-proto/gram-test/circles-in-triangle.jl
@@ -58,9 +58,9 @@ gram = sparse(J, K, values)
 guess = hcat(
   Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
   Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)),
-  Engine.plane(BigFloat[1, 0, 0], BigFloat(1)),
-  Engine.plane(BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(1)),
-  Engine.plane(BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(1)),
+  Engine.plane(-BigFloat[1, 0, 0], BigFloat(-1)),
+  Engine.plane(-BigFloat[cos(2pi/3), sin(2pi/3), 0], BigFloat(-1)),
+  Engine.plane(-BigFloat[cos(-2pi/3), sin(-2pi/3), 0], BigFloat(-1)),
   Engine.sphere(BigFloat[-1, 0, 0], BigFloat(1//5)),
   Engine.sphere(BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//5)),
   Engine.sphere(BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//5)),
diff --git a/notes/inversive.md b/notes/inversive.md
index 6de7ef2..9cb2e19 100644
--- a/notes/inversive.md
+++ b/notes/inversive.md
@@ -7,7 +7,7 @@ These coordinates are of form $I=(c, r, x, y, z)$ where we think of $c$ as the c
 | Entity or Relationship                                                         | Representation                                                                                                                                                                                                                                                                                                                                                                                                                       | Comments/questions                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                       |
 | ------------------------------------------------------------------------------ | ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ | ---------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
 | Sphere s with radius r>0 centered on P = (x,y,z)                               | $I_s = (1/c, 1/r, x/r, y/r, z/r)$ satisfying $Q(I_s,I_s) = -1$, i.e., $c = r/(\|P\|^2 - r^2)$.                                                                                                                                                                                                                                                                                                                                       | Can also write $I_s = (\|P\|^2/r - r, 1/r, x/r. y/r, z/r)$ -- so there is no trouble if $\|E_{I_s}\| = r$, just get first coordinate to be 0.                                                                                                                                                                                                                                                                                                                                                                                                                                            |
-| Plane p with unit normal (x,y,z), a distance s from origin                     | $I_p = (2s, 0, x, y, z)$                                                                                                                                                                                                                                                                                                                                                                                                             | Note $Q(I_p, I_p)$ is still -1. Also, there are two representations for each plane through the origin, namely $(0,0,x,y,z)$ and $(0,0,-x,-y,-z)$                                                                                                                                                                                                                                                                                                                                                                                                                                         |
+| Plane p with unit normal (x,y,z) through the point s(x,y,z)                    | $I_p = (-2s, 0, -x, -y, -z)$                                                                                                                                                                                                                                                                                                                                                                                                         | Note $Q(I_p, I_p)$ is still −1.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |
 | Point P with Euclidean coordinates (x,y,z)                                     | $I_P = (\|P\|^2, 1, x, y, z)$                                                                                                                                                                                                                                                                                                                                                                                                        | Note $Q(I_P,I_P) = 0$.  Because of this we might choose  some other scaling of the inversive coordinates, say $(\||P\||,1/\||P\||,x/\||P\||,y/\||P\||,z/\||P\||)$ instead, but that fails at the origin, and likely won't have some of the other nice properties listed below.  Note that scaling just the co-radius by $s$ and the radius by $1/s$ (which still preserves $Q=0$) dilates by a factor of $s$ about the origin, so that $(\|P\|, \|P\|, x, y, z)$, which might look symmetric, would actually have to represent the Euclidean point $(x/\||P\||, y/\||P\||, z/\||P\||)$ . |
 | ∞, the "point at infinity"                                                     | $I_\infty = (1,0,0,0,0)$                                                                                                                                                                                                                                                                                                                                                                                                             | The only solution to $Q(I,I) = 0$ not covered by the above case.                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                         |
 | P lies on sphere or plane given by I                                           | $Q(I_P, I) = 0$                                                                                                                                                                                                                                                                                                                                                                                                                      |                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                          |