diff --git a/content/http..mathhub.info/FrameIT/frameworld/$Frameworld$Meta.omdoc.xz b/content/http..mathhub.info/FrameIT/frameworld/$Frameworld$Meta.omdoc.xz
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Binary files a/content/http..mathhub.info/FrameIT/frameworld/$Frameworld$Meta.omdoc.xz and b/content/http..mathhub.info/FrameIT/frameworld/$Frameworld$Meta.omdoc.xz differ
diff --git a/narration/MetaTheories.omdoc b/narration/MetaTheories.omdoc
index 34be72d83c4920796d703191a93355e36eccee11..ff67f80de510a6fbfa551bc01cc1b8f9f1c8ca83 100644
--- a/narration/MetaTheories.omdoc
+++ b/narration/MetaTheories.omdoc
@@ -1,2 +1,2 @@
 <?xml version="1.0" encoding="UTF-8"?>
-<omdoc base="http://mathhub.info/FrameIT/frameworld/MetaTheories.omdoc"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#0.0.0:20521.388.0"/><meta property="http://cds.omdoc.org/?metadata?importedby"><om:OMOBJ xmlns:om="http://www.openmath.org/OpenMath"><om:OMLIT type="http://www.openmath.org/cd?OpenMath?OMSTR" value="mmt-omdoc"/></om:OMOBJ></meta></metadata><instruction text="namespace http://mathhub.info/FrameIT/frameworld"/><instruction text="fixmeta http://cds.omdoc.org/urtheories?LF"/><opaque format="text"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#125.5.0:253.5.128"/></metadata>MMT meta keys and value constructors for meta annotations of facts in situation theories and (input/output) facts in scrolls</opaque><mref name="[http://mathhub.info/FrameIT/frameworld?MetaAnnotations]" target="http://mathhub.info/FrameIT/frameworld?MetaAnnotations"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#255.6.0:276.6.21"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?2DPoints]" target="http://mathhub.info/FrameIT/frameworld?2DPoints"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#685.24.1:699.24.15"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITBasics]" target="http://mathhub.info/FrameIT/frameworld?FrameITBasics"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#1094.48.0:1113.48.19"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITRectangles]" target="http://mathhub.info/FrameIT/frameworld?FrameITRectangles"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#4340.108.0:4363.108.23"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITTriangles]" target="http://mathhub.info/FrameIT/frameworld?FrameITTriangles"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#7437.158.0:7459.158.22"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITCircle]" target="http://mathhub.info/FrameIT/frameworld?FrameITCircle"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#10771.202.0:10790.202.19"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITCone]" target="http://mathhub.info/FrameIT/frameworld?FrameITCone"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#12754.240.0:12771.240.17"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITCylinder]" target="http://mathhub.info/FrameIT/frameworld?FrameITCylinder"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#16111.295.0:16132.295.21"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITTheories]" target="http://mathhub.info/FrameIT/frameworld?FrameITTheories"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#17019.318.0:17040.318.21"/></metadata></mref><opaque format="text"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#19961.370.0:20056.370.95"/></metadata>The meta theory to use for situation theories, scroll problem, and scroll solution theories</opaque><mref name="[http://mathhub.info/FrameIT/frameworld?FrameworldMeta]" target="http://mathhub.info/FrameIT/frameworld?FrameworldMeta"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#20058.371.0:20078.371.20"/></metadata></mref></omdoc>
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+<omdoc base="http://mathhub.info/FrameIT/frameworld/MetaTheories.omdoc"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#0.0.0:20627.388.0"/><meta property="http://cds.omdoc.org/?metadata?importedby"><om:OMOBJ xmlns:om="http://www.openmath.org/OpenMath"><om:OMLIT type="http://www.openmath.org/cd?OpenMath?OMSTR" value="mmt-omdoc"/></om:OMOBJ></meta></metadata><instruction text="namespace http://mathhub.info/FrameIT/frameworld"/><instruction text="fixmeta http://cds.omdoc.org/urtheories?LF"/><opaque format="text"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#125.5.0:253.5.128"/></metadata>MMT meta keys and value constructors for meta annotations of facts in situation theories and (input/output) facts in scrolls</opaque><mref name="[http://mathhub.info/FrameIT/frameworld?MetaAnnotations]" target="http://mathhub.info/FrameIT/frameworld?MetaAnnotations"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#255.6.0:276.6.21"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?2DPoints]" target="http://mathhub.info/FrameIT/frameworld?2DPoints"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#685.24.1:699.24.15"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITBasics]" target="http://mathhub.info/FrameIT/frameworld?FrameITBasics"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#1094.48.0:1113.48.19"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITRectangles]" target="http://mathhub.info/FrameIT/frameworld?FrameITRectangles"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#4442.108.0:4465.108.23"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITTriangles]" target="http://mathhub.info/FrameIT/frameworld?FrameITTriangles"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#7539.158.0:7561.158.22"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITCircle]" target="http://mathhub.info/FrameIT/frameworld?FrameITCircle"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#10873.202.0:10892.202.19"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITCone]" target="http://mathhub.info/FrameIT/frameworld?FrameITCone"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#12860.240.0:12877.240.17"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITCylinder]" target="http://mathhub.info/FrameIT/frameworld?FrameITCylinder"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#16217.295.0:16238.295.21"/></metadata></mref><mref name="[http://mathhub.info/FrameIT/frameworld?FrameITTheories]" target="http://mathhub.info/FrameIT/frameworld?FrameITTheories"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#17125.318.0:17146.318.21"/></metadata></mref><opaque format="text"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#20067.370.0:20162.370.95"/></metadata>The meta theory to use for situation theories, scroll problem, and scroll solution theories</opaque><mref name="[http://mathhub.info/FrameIT/frameworld?FrameworldMeta]" target="http://mathhub.info/FrameIT/frameworld?FrameworldMeta"><metadata><link rel="http://cds.omdoc.org/mmt?metadata?sourceRef" resource="http://mathhub.info/FrameIT/frameworld/MetaTheories.mmt#20164.371.0:20184.371.20"/></metadata></mref></omdoc>
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diff --git a/source/MetaTheories.mmt b/source/MetaTheories.mmt
index 39ae590e5d6432951e2e52ef98b4fa5c8a2e0501..4d2c522119aae85c7998f5b8c7ced67dcfa1ac25 100644
--- a/source/MetaTheories.mmt
+++ b/source/MetaTheories.mmt
@@ -88,10 +88,10 @@ theory FrameITBasics =
        // projection and distance function for planes ❙
        helperFunctionForProjection : plane ⟶ point ⟶ ℝ ❘ = [e, p] ( < ( Pnormal e )  , ( planeBase e ) > - < ( Pnormal e ) , p > ) / ( < ( Pnormal e ) , ( Pnormal e ) >  )  ❘ # helperT 1 2 prec 1 ❙
        orthogonalProjectionPointToPlane : plane ⟶ point ⟶ point ❘ = [e,p] p p+p ( ( helperT e p ) rdotp( Pnormal e ) ) ❘ # orthogProjPointToPlane 1 2 prec 1 ❙
-       distancePlanePoint : plane ⟶ point ⟶ ℝ ❘ = [e,p] ( doAbs ( < Pnormal e , p > - < Pnormal e  , Pbase e > ) ) / ( |   Pnormal  e  |  )  ❘ # distPlP 1 2 prec 1 ❙
+       distancePlanePoint : plane ⟶ point ⟶ ℝ ❘ = [e,p] ( doAbs ( < Pnormal e , p > - < Pnormal e  , Pbase e > ) ) / ( |   Pnormal  e  | + epsi  )  ❘ # distPlP 1 2 prec 1 ❙
 
        // functions to calculate the angle between a plane and a line ❙
-       anglePlaneLineCalc : point ⟶ point ⟶ ℝ ❘ = [v,w] asin ( < v,w > / (|v| ⋅ |w|)) ❘ # plcalc 1 2 prec 1 ❙
+       anglePlaneLineCalc : point ⟶ point ⟶ ℝ ❘ = [v,w] asin ( < v,w > / ( (|v| ⋅ |w|) + epsi  )) ❘ # plcalc 1 2 prec 1 ❙
        anglePlaneLine : plane ⟶ line ⟶ ℝ ❘ = [e,l] plcalc ( Pnormal e ) ( directionL l ) ❘ # ∠pl 1 2 prec 1 ❙
        radToDegree : ℝ ⟶ ℝ ❘ = [r] ( r / pi_num ) ⋅ 180.0  ❘ # radToDegree 1 prec 1 ❙
 
@@ -100,7 +100,7 @@ theory FrameITBasics =
 
        lineOnProofFrom: {l } ⊦ ( from l )  on l ❘ # fromOnProofLine 1  ❙
        lineOnProofTo: {l } ⊦ ( to l )  on l ❘ # toOnProofLine 1  ❙
-
+       normalizeSafe: point ⟶ point ❘ = [v] (1.0 / ( | v | + epsi )) ⋅⋅ v ❘ # normS 1 ❙
 
 
  ❚
@@ -228,7 +228,7 @@ theory FrameITCircle =
       angleCircleLine : circle ⟶ line ⟶ ℝ ❘ = [c,l] radToDegree (  plcalc ( Pnormal ( planeCircle c ) )       ( directionL l )    ) ❘ # ∠cl 1 2 prec 1 ❙
       orthogonalCircleLine : circle ⟶ line ⟶ bool ❘ # orthogCL 1 2 ❙
 
-      pointsToPlaneNormal : point ⟶ point ⟶ point ⟶ plane ❘ = [p1,p2,p3] Ppn p1 ( normalize ( (p2 p-p p1 ) Vcross ( p3 p-p p1) ) ) ❘ # pToPlane 1 2 3 ❙
+      pointsToPlaneNormal : point ⟶ point ⟶ point ⟶ plane ❘ = [p1,p2,p3] Ppn p1 ( normalizeSafe ( (p2 p-p p1 ) Vcross ( p3 p-p p1) ) ) ❘ # pToPlane 1 2 3 ❙
       wrap3EdgePointsToPlaneParametric : point ⟶ point ⟶ point ⟶ plane ❘ = [p1,p2,p3 ] Ppara p1 (p2 p-p p1 ) ( p3 p-p p1) ❘ # pointsToPlanePara 1 2 3 ❙
 
       parallelCircles: circle ⟶ circle ⟶ bool ❘ # paraCirc 1 2 ❙