Fact.cs

        if (df == null)
            return null;

        Scroll.ScrollSymbolFact scoll_symbol_fact = (Scroll.ScrollSymbolFact)fact;

        // get the radius
        float radius = ((OMF)((OMA)scoll_symbol_fact.df).arguments[2]).f;

        OMA planeOMA = (OMA)((OMA)scoll_symbol_fact.df).arguments[0];
        string planeApplicant = ((OMS)planeOMA.applicant).uri;

        string A_uri;
        Vector3 normal;
        // Getting the plane
        // IN case of a normale plane
        if (planeApplicant.Equals(MMT_OMS_URI.pointNormalPlane))
        {
            //OMA pointAOMA = (OMA)planeOMA.arguments[0];
            A_uri = ParsingDictionary.parseTermsToId[planeOMA.arguments[0].ToString()];

            OMA n = (OMA)planeOMA.arguments[1];
            normal = new Vector3(((OMF)n.arguments[0]).f, ((OMF)n.arguments[1]).f, ((OMF)n.arguments[2]).f);
        }
        // In case of parametrized plane
        else if (planeApplicant.Equals(MMT_OMS_URI.ParametrizedPlane))
        {
            A_uri = ParsingDictionary.parseTermsToId[planeOMA.arguments[0].ToString()];

            OMA vOMA = (OMA)planeOMA.arguments[1];
            OMA wOMA = (OMA)planeOMA.arguments[2];

            Vector3 v = new Vector3(((OMF)vOMA.arguments[0]).f, ((OMF)vOMA.arguments[1]).f, ((OMF)vOMA.arguments[2]).f);
            Vector3 w = new Vector3(((OMF)wOMA.arguments[0]).f, ((OMF)wOMA.arguments[1]).f, ((OMF)wOMA.arguments[2]).f);

            normal = Vector3.Cross(v, w).normalized;
        }
        // incase of smth else. Shouldn't hapepen unless there is an error
        else throw new ArgumentException("Invalid planeApplicant: " + planeApplicant);

        // get the mid point uri
        string parse_id_M = ((OMA)scoll_symbol_fact.df).arguments[1].ToString();
        string M_uri = ParsingDictionary.parseTermsToId[parse_id_M];

        if (StageStatic.stage.factState.ContainsKey(M_uri)
         && StageStatic.stage.factState.ContainsKey(A_uri))
            return new CircleFact(M_uri, A_uri, radius, normal, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "○" + Point1.Label;

    /// <summary>
    /// Constructs struct for right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="p1URI"> <see cref="Pid1"/></param>
    /// <param name="p2URI"> <see cref="Pid2"/></param>
    /// <param name="radius"> <see cref="radius"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateCircleFactDeclaration(string p1URI, string p2URI, float radius, Vector3 normal)
    {
        List<SOMDoc> outerArguments = new List<SOMDoc>
        {
           //CirclePlane,
           new OMA(
               //PointNormalPlane,
               new OMS(MMT_OMS_URI.pointNormalPlane), 
               //planeArgs,
               new List<SOMDoc> {
                    //base point of the circle plane?,
                    new OMS(p2URI),
                    //NormalVector,
                    new OMA(
                        //"Vector"
                        new OMS(MMT_OMS_URI.Tuple),
                        //normalArgs,
                        new List<SOMDoc> {
                            new OMF(normal.x),
                            new OMF(normal.y),
                            new OMF(normal.z)
                        }
                    ),
                }
            ),
           //middlePoint,
           new OMS(p1URI),
           //Radius,
           new OMF(radius),
        };

        // Do i need this here? doubt 
        SOMDoc tp = new OMS(MMT_OMS_URI.CircleType3d);
        SOMDoc df = new OMA(new OMS(MMT_OMS_URI.MkCircle3d), outerArguments);

        MMTSymbolDeclaration mmtDecl = new MMTSymbolDeclaration(Label, tp, df);
        AddFactResponse.sendAdd(mmtDecl, out _URI);

        return mmtDecl;
    }

    /// \copydoc Fact.hasDependentFacts
    public override bool hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Pid1, Pid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Point1.Label;

        // obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = _Facts[this.Lid2].Label;

        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(CircleFact f1, CircleFact f2) 
        => DependentFactsEquivalent(f1, f2)
        && Math3d.IsApproximatelyEqual(f1.normal, f2.normal)
        && f1.radius.IsApproximatelyEqual(f2.radius);
}

/// <summary>
/// A <see cref="PointFact"/> on a <see cref="CircleFact"/>
/// </summary>
public class OnCircleFact : FactWrappedCRTP<OnCircleFact>
{
    /// <summary> the point on the circle  </summary>
    public string Pid;
    /// <summary> the circle, which the point is on  </summary>
    public string Cid;

    [JsonIgnore]
    public PointFact Point { get => (PointFact)FactOrganizer.AllFacts[Pid]; }
    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactOrganizer.AllFacts[Cid]; }

    /// <summary> \copydoc Fact.Fact </summary>
    public OnCircleFact() : base()
    {
        this.Pid = null;
        this.Cid = null;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public OnCircleFact(OnCircleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Pid], old_to_new[fact.Cid]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="pid">sets <see cref="Pid"/></param>
    /// <param name="cid">sets <see cref="Cid"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OnCircleFact(string pid, string cid, FactOrganizer organizer) : base(organizer)
        => init(pid, cid);

    /// <summary>
    /// Initiates <see cref="Pid"/>, <see cref="Rid"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid">sets <see cref="Pid"/></param>
    /// <param name="cid">sets <see cref="Cid"/></param>
    private void init(string pid, string cid)
    {
        this.Pid = pid;
        this.Cid = cid; ;

        RecalulateTransform();

        SOMDoc tp =
            new OMA(
                new OMS(MMT_OMS_URI.Ded),
                new List<SOMDoc> {
                    new OMA(
                        new OMS(MMT_OMS_URI.OnCircle),
                        new List<SOMDoc> {
                            new OMS(cid),
                            new OMS(pid),
        }),});

        SOMDoc df = null;

        AddFactResponse.sendAdd(new MMTSymbolDeclaration(Label, tp, df), out _URI);
    }

    protected override void RecalulateTransform()
    {
        Position = Point.Position;
        Rotation = Circle.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="pid">sets <see cref="Pid"/></param>
    /// <param name="cid">sets <see cref="Cid"/></param>
    /// <param name="uri">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OnCircleFact(string pid, string cid, string uri, FactOrganizer organizer) : base(organizer)
    {
        this.Pid = pid;
        this.Cid = cid;
        this._URI = uri;
        _ = this.Label;

        RecalulateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static OnCircleFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;

        OMA tp = (OMA)((Scroll.ScrollSymbolFact)fact).tp;
        if (tp == null)
            return null;

        string circleUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[0]).uri;
        string pointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollSymbolFact)fact).tp).arguments[0]).arguments[1]).uri;

        if (StageStatic.stage.factState.ContainsKey(pointUri)
         && StageStatic.stage.factState.ContainsKey(circleUri))
            return new OnCircleFact(pointUri, circleUri, uri, StageStatic.stage.factState);

        //If dependent facts do not exist return null
        else
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => Point.Label + "∈" + Circle.Label;

    /// \copydoc Fact.hasDependentFacts
    public override bool hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Pid, Cid };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Point.Label + "∈" + Circle.Label;
        obj.GetComponent<FactWrapper>().fact = this;

        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(OnCircleFact c1, OnCircleFact c2)
        => DependentFactsEquivalent(c1, c2);
}

/// <summary>
/// Angle comprised of a line and a circle 
/// </summary>
public class AngleCircleLineFact : FactWrappedCRTP<AngleCircleLineFact>
{
    /// @{ <summary>
    /// One <see cref="Fact.Id">Id</see> of a <see cref="RayFact">RayFact</see> and a <see cref="CircleFact">CircleFact</see>  defining Angle [<see cref="Cid1"/>, <see cref="Rid2"/>].
    /// </summary>
    public string Cid1, Rid2;
    /// @}

    [JsonIgnore]
    public AbstractLineFact Ray { get => (AbstractLineFact)FactOrganizer.AllFacts[Rid2]; }
    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    //TODO? deg or rad?
    [JsonIgnore]
    public float angle;

    [JsonIgnore]
    public Vector3 intersection;

    /// <summary> \copydoc Fact.Fact </summary>
    public AngleCircleLineFact() : base()
    {
        this.Cid1 = null;
        this.Rid2 = null;
        this.angle = 0.0f;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public AngleCircleLineFact(AngleCircleLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1], old_to_new[fact.Rid2]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="rid2">sets <see cref="Rid2"/></param>
    /// <param name="angle"> sets the angle </param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AngleCircleLineFact(string cid1, string rid2, FactOrganizer organizer) : base(organizer)
        => init(cid1, rid2);

    /// <summary>
    /// Initiates <see cref="Cid1"/>, <see cref="Rid2"/>, <see cref="angle"/> <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="rid2">sets <see cref="Rid2"/></param>
    /// <param name="angle"> sets the angle </param>
    private void init(string cid1, string rid2)
    {
        this.Cid1 = cid1;
        this.Rid2 = rid2;
        this.angle = Math3d.AngleVectorPlane(Ray.Dir, Circle.normal).ToDegrees();
        Math3d.LinePlaneIntersection(out intersection, Ray.Point1.Position, Ray.Dir, Circle.normal, Circle.Position);

        RecalulateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(angle, cid1, rid2);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalulateTransform()
    {
        Position = intersection;
        { //Rotation
            Vector3 from = (Circle.Position - Position).normalized;

            Vector3 angle_to = Math3d.IsApproximatelyEqual(intersection, Ray.Point1.Position)
                ? Ray.Point2.Position
                : Ray.Point1.Position;
            Vector3 to = (angle_to - Position).normalized;

            Vector3 up = Vector3.Cross(to, from);
            Vector3 forwoard = (from + to).normalized;

            if (up.sqrMagnitude < Math3d.vectorPrecission)
            { //Angle is 180° (or 0°)
                Vector3 arbitary = up.normalized == Vector3.forward ? Vector3.right : Vector3.forward;
                up = Vector3.Cross(arbitary, to);
                forwoard = Vector3.Cross(up, to);
            }

            Rotation = Quaternion.LookRotation(forwoard, up);
        }
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="´Rid2">sets <see cref="Rid2"/></param>
    /// <param name="angle"> sets the angle </param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AngleCircleLineFact(string Cid1, string Rid2, float angle, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Rid2 = Rid2;
        this.angle = angle;

        this._URI = backendURI;
        _ = this.Label;

        RecalulateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static AngleCircleLineFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        string
            CircleUri,
            RayUri;

        OMA df = (OMA)((Scroll.ScrollValueFact)fact).lhs;
        if (df == null)
            return null;

        // init it with 0 degrees, so we don't accidentally generate orthogonalfacts 
        // and the parsing works correctly if smb ever adds a scroll for this
        float angle = 0.0f;

        if ((((Scroll.ScrollValueFact)fact).value) != null)
            angle = ((OMF)(((Scroll.ScrollValueFact)fact).value)).f;


        CircleUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;
        RayUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[1]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri)
         && StageStatic.stage.factState.ContainsKey(RayUri))

            return new AngleCircleLineFact(CircleUri, RayUri, angle, uri, StageStatic.stage.factState);
        else return null; //If dependent facts do not exist return null
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "∠" + Circle.Label + Ray.Label;

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="val">Angle != 90f, _not checked_</param>
    /// <param name="c1URI"><see cref="Cid1"/></param>
    /// <param name="R2URI"><see cref="Rid2"/></param>
    /// <param name="val"><see cref="angle"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(float val, string c1URI, string r2URI)
    {
        SOMDoc lhs =
            new OMA(
                new OMS(MMT_OMS_URI.AnglePlaneLine),
                new List<SOMDoc> {
                    new OMS(c1URI),
                    new OMS(r2URI),
                }
            );

        SOMDoc valueTp = new OMS(MMT_OMS_URI.RealLit);
        SOMDoc value = new OMF(val);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Rid2 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle.Label;
        obj.transform.GetChild(1).gameObject.GetComponent<TextMeshProUGUI>().text = Ray.Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(AngleCircleLineFact f1, AngleCircleLineFact f2)
        => DependentFactsEquivalent(f1, f2);
}

/// <summary>
/// A RadiusFact that corresponds to a <see cref="CircleFact">PointFacts</see> and has a float value (the actual radius).
/// </summary>
public class RadiusFact : FactWrappedCRTP<RadiusFact>
{
    /// <summary> The circle corresponding to the radius </summary>
    public string Cid1;

    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    /// <summary> The radius as a float </summary>
    public float rad;

    /// <summary> \copydoc Fact.Fact </summary>
    public RadiusFact() : base()
    {
        this.Cid1 = null;
        this.rad = 0.0f;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public RadiusFact(RadiusFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public RadiusFact(string cid1, FactOrganizer organizer) : base(organizer)
        => init(cid1);

    /// <summary>
    /// Initiates <see cref="Cid1"/> and <see cref="rad"/>
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    private void init(string cid1)
    {
        this.Cid1 = cid1;
        this.rad = Circle.radius;

        RecalulateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, this.rad);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalulateTransform()
    {
        Position = Circle.Position;
        Rotation = Circle.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public RadiusFact(string Cid1, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;

        this._URI = backendURI;
        _ = this.Label;

        RecalulateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static RadiusFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        string CircleUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri))

            return new RadiusFact(CircleUri, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "r " + Circle.Label;

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="rad"> see <see cref="rad"/></param>
    /// <param name="c1URI"> see <see cref="Cid1"/></param>

    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, float rad)
    {
        SOMDoc lhs =
            new OMA(
                new OMS(MMT_OMS_URI.RadiusCircleMetric),
                new List<SOMDoc> {
                    new OMS(c1URI),
                }
            );

        SOMDoc valueTp = new OMS(MMT_OMS_URI.RealLit);
        SOMDoc value = new OMF(rad);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override bool hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = "r: " + Circle.Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(RadiusFact f1, RadiusFact f2)
        => DependentFactsEquivalent(f1, f2);
}

/// <summary>
/// Area of a <see cref="CircleFact">CircleFact</see> 
/// </summary>
public class AreaCircleFact : FactWrappedCRTP<AreaCircleFact>
{
    /// <summary> the circle <see cref="CircleFact">CircleFact</see>  </summary>
    public string Cid1;

    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    /// <summary> the area which is contained by the circle </summary>
    public float A;


    /// <summary> \copydoc Fact.Fact </summary>
    public AreaCircleFact() : base()
    {
        this.Cid1 = null;
        this.A = 0.0f;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public AreaCircleFact(AreaCircleFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AreaCircleFact(string cid1, FactOrganizer organizer) : base(organizer)
        => init(cid1);

    /// <summary>
    /// Initiates <see cref="Cid1"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>

    private void init(string cid1)
    {
        this.Cid1 = cid1;

        this.A = Circle.radius * Circle.radius * ((float)System.Math.PI);

        RecalulateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, this.A);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalulateTransform()
    {
        Position = Circle.Position;
        Rotation = Circle.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public AreaCircleFact(string Cid1, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;

        this._URI = backendURI;
        _ = this.Label;

        RecalulateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static AreaCircleFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;
        string CircleUri = ((OMS)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).uri;

        if (StageStatic.stage.factState.ContainsKey(CircleUri))
            return new AreaCircleFact(CircleUri, uri, StageStatic.stage.factState);
        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "A(" + Circle.Label + ")";

    /// <summary>
    /// Constructs a response, that is sent to the MMT-Server
    /// </summary>
    /// <param name="area"> area of the circle </param>
    /// <param name="c1URI">  <see cref="Cid1"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, float area)
    {
        SOMDoc lhs =
            new OMA(
                new OMS(MMT_OMS_URI.AreaCircle),
                new List<SOMDoc> {
                    new OMS(c1URI),
                }
            );

        SOMDoc valueTp = new OMS(MMT_OMS_URI.RealLit);
        SOMDoc value = new OMF(area);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle.Label;
        obj.GetComponent<FactWrapper>().fact = this;
        return obj;
    }

    /// \copydoc Fact.GetHashCode
    /// is this a problem?
    public override int GetHashCode()
        => base.GetHashCode() ^ A.GetHashCode();

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(AreaCircleFact f1, AreaCircleFact f2)
        => DependentFactsEquivalent(f1, f2);
}

/// <summary>
/// The volume of a cone A  defined by a base area  <see cref="CircleFact">CircleFact</see>, an apex <see cref="PointFact">PointFact</see> and the volume as float
/// </summary>
public class ConeVolumeFact : FactWrappedCRTP<ConeVolumeFact>
{
    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    ///  <summary> a <see cref="PointFact">PointFact</see> describing the apex point  </summary>
    public string Pid1;
    [JsonIgnore]
    public PointFact Point { get => (PointFact)FactOrganizer.AllFacts[Pid1]; }

    ///  <summary> the volume of the cone as a float </summary>
    public float vol;

    /// <summary> \copydoc Fact.Fact </summary>
    public ConeVolumeFact() : base()
    {
        this.Cid1 = null;
        this.Pid1 = null;
        this.vol = 0.0f;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public ConeVolumeFact(ConeVolumeFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1], old_to_new[fact.Pid1], fact.vol);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public ConeVolumeFact(string cid1, string pid1, float vol, FactOrganizer organizer) : base(organizer)
        => init(cid1, pid1, vol);

    /// <summary>
    /// Initiates <see cref="Cid1"/>, <see cref="Pid1"/>, <see cref="vol"/>,  <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="pid1">sets <see cref="Pid1"/></param>
    /// <param name="vol">sets <see cref="vol"/></param>
    private void init(string cid1, string pid1, float vol)
    {
        this.Cid1 = cid1;
        this.Pid1 = pid1;
        this.vol = vol;

        RecalulateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, pid1, vol);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalulateTransform()
    {
        Position = Point.Position;
        Rotation = Circle.Rotation;
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Pid1">sets <see cref="Pid1"/></param>
    /// <param name="volume">sets <see cref="vol"/></param>
    /// <param name="backendURI">MMT URI</param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public ConeVolumeFact(string Cid1, string Pid1, float volume, string backendURI, FactOrganizer organizer) : base(organizer)
    {
        this.Cid1 = Cid1;
        this.Pid1 = Pid1;
        this.vol = volume;

        this._URI = backendURI;
        _ = this.Label;

        RecalulateTransform();
    }

    /// \copydoc Fact.parseFact(Scroll.ScrollFact)
    public new static ConeVolumeFact parseFact(Scroll.ScrollFact fact)
    {
        string uri = fact.@ref.uri;

        if (((Scroll.ScrollValueFact)fact).lhs == null)
            return null;

        string CircleUri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[0]).uri;
        string PointUri = ((OMS)((OMA)((OMA)((Scroll.ScrollValueFact)fact).lhs).arguments[0]).arguments[1]).uri;
        float volume = 0.0f;
        if ((((Scroll.ScrollValueFact)fact).value) != null)
            volume = ((OMF)((Scroll.ScrollValueFact)fact).value).f;

        if (StageStatic.stage.factState.ContainsKey(CircleUri) && StageStatic.stage.factState.ContainsKey(PointUri))

            return new ConeVolumeFact(CircleUri, PointUri, volume, uri, StageStatic.stage.factState);

        else    //If dependent facts do not exist return null
            return null;
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel()
        => "V(" + Circle.Label + "," + Point.Label + ")";

    /// <summary>
    /// Constructs struct for not-right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="c1URI"> Uri for <see cref="Cid1"/></param>
    /// <param name="p1URI"> Uri for <see cref="Pid1"/></param>
    /// <param name="val"> <see cref="vol"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    private MMTDeclaration generateMMTDeclaration(string c1URI, string p1URI, float val)
    {
        SOMDoc lhs =
            new OMA(
                new OMS(MMT_OMS_URI.VolumeCone),

                new List<SOMDoc> {
                    new OMA(new OMS(MMT_OMS_URI.ConeOfCircleApex),
                        new List<SOMDoc> {
                            new OMS(c1URI),
                            new OMS(p1URI),
                         }
                    ),
                }
            );

        SOMDoc valueTp = new OMS(MMT_OMS_URI.RealLit);
        SOMDoc value = new OMF(val);

        return new MMTValueDeclaration(this.Label, lhs, valueTp, value);
    }

    /// \copydoc Fact.hasDependentFacts
    public override Boolean hasDependentFacts()
        => true;

    /// \copydoc Fact.getDependentFactIds
    public override string[] getDependentFactIds()
        => new string[] { Cid1, Pid1 };

    /// \copydoc Fact.instantiateDisplay(GameObject, Transform)
    public override GameObject instantiateDisplay(GameObject prefab, Transform transform)
    {
        var obj = GameObject.Instantiate(prefab, Vector3.zero, Quaternion.identity, transform);
        obj.transform.GetChild(0).gameObject.GetComponent<TextMeshProUGUI>().text = Circle.Label + Point.Label;
        obj.GetComponent<FactWrapper>().fact = this;

        return obj;
    }

    /// \copydoc Fact.Equivalent(Fact, Fact)
    protected override bool EquivalentWrapped(ConeVolumeFact f1, ConeVolumeFact f2)
        => DependentFactsEquivalent(f1, f2);
}

/// <summary>
/// The fact that the plane of a <see cref="CircleFact">CircleFact</see> and the line <see cref="RayFact>RayFact</see> are orthogonal
/// </summary>
public class OrthogonalCircleLineFact : FactWrappedCRTP<OrthogonalCircleLineFact>
{
    ///  <summary> a <see cref="CircleFact">CircleFact</see> describing the base area </summary>
    public string Cid1;
    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactOrganizer.AllFacts[Cid1]; }

    ///  <summary> a <see cref="RayFact">Rayfact</see> describing the line </summary>
    public string Lid1;
    [JsonIgnore]
    public AbstractLineFact Ray { get => (AbstractLineFact)FactOrganizer.AllFacts[Lid1]; }

    //TODO? deg or rad?
    [JsonIgnore]
    public float angle = 90f;

    [JsonIgnore]
    public Vector3 intersection;


    /// <summary> \copydoc Fact.Fact </summary>
    public OrthogonalCircleLineFact() : base()
    {
        this.Cid1 = null;
        this.Lid1 = null;
    }

    /// <summary>
    /// Copies <paramref name="fact"/> by initiating new MMT %Fact.
    /// </summary>
    /// <param name="fact">Fact to be copied</param>
    /// <param name="old_to_new"><c>Dictionary</c> mapping <paramref name="fact"/>.<see cref="getDependentFactIds"/> in <paramref name="fact"/>.<see cref="Fact._Facts"/> to corresponding <see cref="Fact.Id"/> in <paramref name="organizer"/> </param>
    /// <param name="organizer">sets <see cref="_Facts"/></param>
    public OrthogonalCircleLineFact(OrthogonalCircleLineFact fact, Dictionary<string, string> old_to_new, FactOrganizer organizer) : base(fact, organizer)
        => init(old_to_new[fact.Cid1], old_to_new[fact.Lid1]);

    /// <summary>
    /// Standard Constructor
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    /// <param name="organizer">sets <see cref="Fact._Facts"/></param>
    public OrthogonalCircleLineFact(string cid1, string lid1, FactOrganizer organizer) : base(organizer)
        => init(cid1, lid1);

    /// <summary>
    /// Initiates <see cref="Pid1"/>, <see cref="Pid2"/>, <see cref="Pid3"/>, <see cref="is_right_angle"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    private void init(string cid1, string lid1)
    {
        this.Cid1 = cid1;
        this.Lid1 = lid1;
        Math3d.LinePlaneIntersection(out intersection, Ray.Point1.Position, Ray.Dir, Circle.normal, Circle.Position);

        RecalulateTransform();

        MMTDeclaration mmtDecl = generateMMTDeclaration(cid1, lid1);
        AddFactResponse.sendAdd(mmtDecl, out this._URI);
    }

    protected override void RecalulateTransform()
    {
        Position = intersection;
        { //Rotation
            Vector3 from = (Circle.Position - Position).normalized;

            Vector3 angle_to = Math3d.IsApproximatelyEqual(intersection, Ray.Point1.Position)
                ? Ray.Point2.Position
                : Ray.Point1.Position;
            Vector3 to = (angle_to - Position).normalized;

            Vector3 up = Vector3.Cross(to, from);
            Vector3 forward = (from + to).normalized;
            Rotation = Quaternion.LookRotation(forward, up);
        }
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Cid1">sets <see cref="Cid1"/></param>
    /// <param name="Lid1">sets <see cref="Lid1"/></param>