CircleFact.cs

using System;
using System.Collections.Generic;
using UnityEngine;
using Newtonsoft.Json;
using REST_JSON_API;
using System.Collections;
using System.Xml.Linq;

/// <summary>
/// A Circle that is made out of a middle point, a plane and a radius
/// </summary>
public class CircleFact : FactWrappedCRTP<CircleFact>
{
    /// <summary> defining the middle point of the circle  </summary>
    public string PidCenter;
    /// <summary>  defining the base point of the circle plane </summary>
    public string PidBase;

    [JsonIgnore]
    public PointFact PointCenter { get => (PointFact)FactRecorder.AllFacts[PidCenter]; }
    [JsonIgnore]
    public PointFact PointBase { get => (PointFact)FactRecorder.AllFacts[PidBase]; }

    /// <summary>  radius of the circle </summary>
    public float radius;
    /// <summary> normal vector of the plane </summary>
    public Vector3 normal;

    /// <summary> \copydoc Fact.Fact </summary>
    public CircleFact() : base()
    {
        this.normal = Vector3.zero;
        this.PidCenter = null;
        this.PidBase = null;
        this.radius = 0;
    }

    /// <summary>
    /// Standard Constructor:
    /// Initiates <see cref="PidCenter"/>, <see cref="PidBase"/>, <see cref="radius"/>,<see cref="dir1"/>,<see cref="dir2"/>, <see cref="Fact._URI"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="pid1">sets <see cref="PidCenter"/></param>
    /// <param name="pid2">sets <see cref="PidBase"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    public CircleFact(string pid1, string pid2, float radius, Vector3 normal) : base()
    {
        this.PidCenter = pid1;
        this.PidBase = pid2;

        this.radius = radius;
        this.normal = normal;
    }

    protected override void RecalculateTransform()
    {
        Position = PointCenter.Position;
        { //Rotation
            Vector3 arbitary_not_normal = normal == Vector3.forward ? Vector3.right : Vector3.forward;
            Vector3 forwoard = Vector3.Cross(arbitary_not_normal, normal);

            Rotation = Quaternion.LookRotation(forwoard, normal);
        }
        LocalScale = new Vector3(radius, 1, radius);
    }

    /// <summary>
    /// Bypasses initialization of new MMT %Fact by using existend URI, _which is not checked for existence_.
    /// </summary>
    /// <param name="Pid1">sets <see cref="PidCenter"/></param>
    /// <param name="Pid2">sets <see cref="PidBase"/></param>
    /// <param name="radius">sets <see cref="radius"/></param>
    /// <param name="normal">sets <see cref="normal"/></param>
    /// <param name="backendURI">MMT URI</param>
    public CircleFact(string Pid1, string Pid2, float radius, Vector3 normal, SOMDoc _ServerDefinition) : base()
    {
        this.PidCenter = Pid1;
        this.PidBase = Pid2;

        this.radius = radius;
        this.normal = normal;

        this.ServerDefinition = _ServerDefinition;
    }

    /// <summary>
    /// parses the Circlefact response of the MMT-Server
    /// </summary>
    /// \copydoc Fact.parseFact(ScrollFact) 
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        if (((MMTGeneralFact)fact).defines is not OMA df)
            yield break;

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

        Vector3 normal;
        // Getting the plane
        // IN case of a normale plane
        if (planeApplicant.Equals(MMTConstants.pointNormalPlane))
        {
            //OMA pointAOMA = (OMA)planeOMA.arguments[0];
            normal = SOMDoc.MakeVector3((OMA)planeOMA.arguments[1]);
        }
        // In case of parametrized plane
        else if (planeApplicant.Equals(MMTConstants.ParametrizedPlane))
        {
            Vector3 v = SOMDoc.MakeVector3((OMA)planeOMA.arguments[1]);
            Vector3 w = SOMDoc.MakeVector3((OMA)planeOMA.arguments[2]);

            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 = df.arguments[1].ToString();
        string M_uri = ParsingDictionary.parseTermsToId[parse_id_M];
        string A_uri = ParsingDictionary.parseTermsToId[planeOMA.arguments[0].ToString()];
        float radius = ((OMLIT<float>)df.arguments[2]).value;

        if (!FactRecorder.AllFacts.ContainsKey(M_uri)
         || !FactRecorder.AllFacts.ContainsKey(A_uri))
            yield break; //If dependent facts do not exist return null

        ret.Add(new CircleFact(M_uri, A_uri, radius, normal, fact.@ref));
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel(FactRecorder name_space)
        => "○" + PointCenter.GetLabel(name_space);

    /// <summary>
    /// Constructs struct for right-angled MMT %Fact <see cref="AddFactResponse"/>
    /// </summary>
    /// <param name="Pid1"> <see cref="PidCenter"/></param>
    /// <param name="p2URI"> <see cref="PidBase"/></param>
    /// <param name="radius"> <see cref="radius"/></param>
    /// <returns>struct for <see cref="AddFactResponse"/></returns>
    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc tp = new OMS(MMTConstants.CircleType3d);

        return new MMTGeneralFact(_LastLabel, tp, Defines());
    }

    public override SOMDoc Defines()
        => new OMA(
            new OMS(MMTConstants.MkCircle3d),
            new SOMDoc[] {
               //CirclePlane,
               new OMA(
                   //PointNormalPlane,
                   new OMS(MMTConstants.pointNormalPlane), 
                   //planeArgs,
                   new SOMDoc[] {
                        //base point of the circle plane?,
                        new OMS(PidBase),
                        //NormalVector,
                        new OMA(
                            //"Vector"
                            new OMS(MMTConstants.Tuple),
                            //normalArgs,
                            new[]  {
                                new OMLIT<float>(normal.x),
                                new OMLIT<float>(normal.y),
                                new OMLIT<float>(normal.z)
                            }
                        ),
                    }
                ),
               //middlePoint,
               new OMS(PidCenter),
               //Radius,
               new OMLIT<float>(radius),
            }
        );

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

    /// \copydoc Fact.getDependentFactIds
    protected override string[] GetDependentFactIds()
        => new string[] { PidCenter, PidBase };

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new CircleFact(old_to_new[this.PidCenter], old_to_new[this.PidBase], this.radius, this.normal);
}

/// <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)FactRecorder.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>
    /// Standard Constructor:
    /// Initiates <see cref="Cid1"/> and <see cref="rad"/>
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    public RadiusFact(string cid1) : base()
    {
        this.Cid1 = cid1;
        this.rad = Circle.radius;
    }

    protected override void RecalculateTransform()
    {
        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>
    public RadiusFact(string Cid1, SOMDoc _ServerDefinition) : base()
    {
        this.Cid1 = Cid1;

        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        string CircleUri = ((OMS)((OMA)((MMTValueFact)fact).lhs).arguments[0]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(CircleUri))
            yield break;

        ret.Add(new RadiusFact(CircleUri, fact.@ref));
    }

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

    /// <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>

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc valueTp = new OMS(MMTConstants.RealLit);
        SOMDoc value = new OMLIT<float>(rad);

        return new MMTValueFact(_LastLabel, Defines(), valueTp, value);
    }

    public override SOMDoc Defines()
        => new OMA(
            new OMS(MMTConstants.RadiusCircleMetric),
            new[] {
                new OMS(Cid1),
            }
        );

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

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

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new RadiusFact(old_to_new[this.Cid1]);
}

/// <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)FactRecorder.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>
    /// Standard Constructor:
    /// Initiates <see cref="Cid1"/> and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    public AreaCircleFact(string cid1) : base()
    {
        this.Cid1 = cid1;
        this.A = Circle.radius * Circle.radius * (float)Math.PI;
    }

    protected override void RecalculateTransform()
    {
        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>
    public AreaCircleFact(string Cid1, SOMDoc _ServerDefinition) : base()
    {
        this.Cid1 = Cid1;

        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        string CircleUri = ((OMS)((OMA)((MMTValueFact)fact).lhs).arguments[0]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(CircleUri))
            yield break;

        ret.Add(new AreaCircleFact(CircleUri, fact.@ref));
    }

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

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc valueTp = new OMS(MMTConstants.RealLit);
        SOMDoc value = new OMLIT<float>(A);

        return new MMTValueFact(_LastLabel, Defines(), valueTp, value);
    }

    public override SOMDoc Defines()
        => new OMA(
            new OMS(MMTConstants.AreaCircle),
            new[] {
                new OMS(Cid1),
            }
        );

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

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

    /// \copydoc Fact.GetHashCode
    public override int GetHashCode()
        => base.GetHashCode() ^ A.GetHashCode();

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new AreaCircleFact(old_to_new[this.Cid1]);
}

/// <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)FactRecorder.AllFacts[Pid]; }
    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactRecorder.AllFacts[Cid]; }

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

    /// <summary>
    /// Standard Constructor:
    /// 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>
    public OnCircleFact(string pid, string cid) : base()
    {
        this.Pid = pid;
        this.Cid = cid;
    }

    protected override void RecalculateTransform()
    {
        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>
    public OnCircleFact(string pid, string cid, SOMDoc _ServerDefinition) : base()
    {
        this.Pid = pid;
        this.Cid = cid;
        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        if (((MMTGeneralFact)fact).type is not OMA type)
            yield break;

        string circleUri = ((OMS)((OMA)type.arguments[0]).arguments[0]).uri;
        string pointUri = ((OMS)((OMA)type.arguments[0]).arguments[1]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(pointUri)
         || !FactRecorder.AllFacts.ContainsKey(circleUri))
            yield break;

        ret.Add(new OnCircleFact(pointUri, circleUri, fact.@ref));
    }

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

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

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

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new OnCircleFact(old_to_new[this.Pid], old_to_new[this.Cid]);

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc tp =
            new OMA(
                new OMS(MMTConstants.Ded),
                new[] {
                    new OMA(
                        new OMS(MMTConstants.OnCircle),
                        new[] {
                            new OMS(Cid),
                            new OMS(Pid),
        }),});

        return new MMTGeneralFact(_LastLabel, tp, Defines());
    }

    public override SOMDoc Defines()
        => null;
}

/// <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)FactRecorder.AllFacts[Rid2]; }
    [JsonIgnore]
    public CircleFact Circle { get => (CircleFact)FactRecorder.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>
    /// Standard Constructor:
    /// 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>
    public AngleCircleLineFact(string cid1, string rid2) : base()
    {
        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);
    }

    protected override void RecalculateTransform()
    {
        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>
    public AngleCircleLineFact(string Cid1, string Rid2, float angle, SOMDoc _ServerDefinition) : base()
    {
        this.Cid1 = Cid1;
        this.Rid2 = Rid2;
        this.angle = angle;

        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        if (((MMTValueFact)fact).lhs is not OMA lhs)
            yield break;

        // 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 (((MMTValueFact)fact).value is OMLIT<float> oMFangle)
            angle = oMFangle.value;

        string CircleUri = ((OMS)lhs.arguments[0]).uri;
        string RayUri = ((OMS)lhs.arguments[1]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(CircleUri)
         || !FactRecorder.AllFacts.ContainsKey(RayUri))
            yield break;

        ret.Add(new AngleCircleLineFact(CircleUri, RayUri, angle, fact.@ref));
    }

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

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc valueTp = new OMS(MMTConstants.RealLit);
        SOMDoc value = new OMLIT<float>(angle);

        return new MMTValueFact(_LastLabel, Defines(), valueTp, value);
    }

    public override SOMDoc Defines()
        => new OMA(
                new OMS(MMTConstants.AnglePlaneLine),
                new[] {
                    new OMS(Cid1),
                    new OMS(Rid2),
                }
            );

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

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

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new AngleCircleLineFact(old_to_new[this.Cid1], old_to_new[this.Rid2]);
}

/// <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)FactRecorder.AllFacts[Cid1]; }

    ///  <summary> a <see cref="RayFact">Rayfact</see> describing the line </summary>
    public string Lid1;
    [JsonIgnore]
    public AbstractLineFact Ray { get => (AbstractLineFact)FactRecorder.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>
    /// Standard Constructor:
    /// Initiates members and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="lid1">sets <see cref="Lid1"/></param>
    public OrthogonalCircleLineFact(string cid1, string lid1) : base()
    {
        this.Cid1 = cid1;
        this.Lid1 = lid1;
        Math3d.LinePlaneIntersection(out intersection, Ray.Point1.Position, Ray.Dir, Circle.normal, Circle.Position);
    }

    protected override void RecalculateTransform()
    {
        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>
    /// <param name="backendURI">MMT URI</param>
    public OrthogonalCircleLineFact(string Cid1, string Lid1, SOMDoc _ServerDefinition) : base()
    {
        this.Cid1 = Cid1;
        this.Lid1 = Lid1;

        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        if (((MMTGeneralFact)fact).type is not OMA type)
            yield break;

        string CircleUri = ((OMS)((OMA)type.arguments[0]).arguments[0]).uri;
        string LineUri = ((OMS)((OMA)type.arguments[0]).arguments[1]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(CircleUri)
         || !FactRecorder.AllFacts.ContainsKey(LineUri))
            yield break;

        ret.Add(new OrthogonalCircleLineFact(CircleUri, LineUri, fact.@ref));
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel(FactRecorder name_space)
        => Circle.GetLabel(name_space) + "⊥" + Ray.GetLabel(name_space);

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc tp = new OMA(
            new OMS(MMTConstants.Ded),
            new[]{
                new OMA(
                    new OMS(MMTConstants.OrthoCircleLine),
                    new[] {
                        new OMS(Cid1),
                        new OMS(Lid1),
                    }
                ),
            }
        );

        return new MMTGeneralFact(_LastLabel, tp, Defines());
    }

    public override SOMDoc Defines()
        => null;

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

    /// \copydoc Fact.getDependentFactIds
    protected override string[] GetDependentFactIds()
        => new string[] { Cid1, Lid1 };

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new OrthogonalCircleLineFact(old_to_new[this.Cid1], old_to_new[this.Lid1]);
}

/// <summary>
/// A fact that describes, that two circles have the same size and is comprised of two <see cref="CircleFact">CircleFacts</see> 
/// </summary>
public class EqualCirclesFact : FactWrappedCRTP<EqualCirclesFact>
{
    /// @{ <summary>
    /// two circles that are meant to be equal in area
    /// </summary>
    public string Cid1, Cid2;
    /// @}

    [JsonIgnore]
    public CircleFact Circle1 { get => (CircleFact)FactRecorder.AllFacts[Cid1]; }
    [JsonIgnore]
    public CircleFact Circle2 { get => (CircleFact)FactRecorder.AllFacts[Cid2]; }

    /// <summary> \copydoc Fact.Fact </summary>
    public EqualCirclesFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
    }

    /// <summary>
    /// Standard Constructor:
    /// Initiates members and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    public EqualCirclesFact(string cid1, string cid2) : base()
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;
    }

    protected override void RecalculateTransform()
    {
        Position = Circle1.Position;
        Rotation = Circle1.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="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="backendURI">MMT URI</param>
    public EqualCirclesFact(string Cid1, string Cid2, SOMDoc _ServerDefinition) : base()
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;

        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        if (((MMTGeneralFact)fact).type is not OMA proof_OMA) // proof DED
            yield break;

        OMA parallel_circles_OMA = (OMA)proof_OMA.arguments[0]; // parallel

        string circleAUri = ((OMS)parallel_circles_OMA.arguments[0]).uri;
        string circleBUri = ((OMS)parallel_circles_OMA.arguments[1]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(circleAUri)
         || !FactRecorder.AllFacts.ContainsKey(circleBUri))
            yield break;

        ret.Add(new EqualCirclesFact(circleAUri, circleBUri, fact.@ref));
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel(FactRecorder name_space)
        => Circle1.GetLabel(name_space) + " ≠ " + Circle2.GetLabel(name_space);

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc tp = new OMA(
            new OMS(MMTConstants.Ded),
            new[] {
                new OMA(
                    new OMS(MMTConstants.EqualityCircles),
                    new[]  {
                        new OMS(Cid1),
                        new OMS(Cid2),
                    }
                )
            }
        );

        return new MMTGeneralFact(_LastLabel, tp, Defines());
    }

    public override SOMDoc Defines()
        => null;

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

    /// \copydoc Fact.getDependentFactIds
    protected override string[] GetDependentFactIds()
        => new string[] { Cid1, Cid2 };

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

    protected override Fact _ReInitializeMe(Dictionary<string, string> old_to_new)
        => new EqualCirclesFact(old_to_new[this.Cid1], old_to_new[this.Cid2]);
}

/// <summary>
/// A fact that describes, that two circles have not the same size and is comprised of two <see cref="CircleFact">CircleFacts</see> 
/// </summary>
public class UnEqualCirclesFact : FactWrappedCRTP<UnEqualCirclesFact>
{
    /// @{ <summary>
    /// two circles that are meant to be unequal in area
    /// </summary>
    public string Cid1, Cid2;
    /// @}

    [JsonIgnore]
    public CircleFact Circle1 { get => (CircleFact)FactRecorder.AllFacts[Cid1]; }
    [JsonIgnore]
    public CircleFact Circle2 { get => (CircleFact)FactRecorder.AllFacts[Cid2]; }

    /// <summary> \copydoc Fact.Fact </summary>
    public UnEqualCirclesFact() : base()
    {
        this.Cid1 = null;
        this.Cid2 = null;
    }

    /// <summary>
    /// Standard Constructor:
    /// Initiates members and creates MMT %Fact Server-Side
    /// </summary>
    /// <param name="cid1">sets <see cref="Cid1"/></param>
    /// <param name="cid2">sets <see cref="Cid2"/></param>
    public UnEqualCirclesFact(string cid1, string cid2) : base()
    {
        this.Cid1 = cid1;
        this.Cid2 = cid2;
    }

    protected override void RecalculateTransform()
    {
        Position = Circle2.Position;
        Rotation = Circle2.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="Cid2">sets <see cref="Cid2"/></param>
    /// <param name="backendURI">MMT URI</param>
    public UnEqualCirclesFact(string Cid1, string Cid2, SOMDoc _ServerDefinition) : base()
    {
        this.Cid1 = Cid1;
        this.Cid2 = Cid2;

        this.ServerDefinition = _ServerDefinition;
    }

    /// \copydoc Fact.parseFact(ScrollFact)
    public new static IEnumerator parseFact(List<Fact> ret, MMTFact fact)
    {
        if (((MMTGeneralFact)fact).type is not OMA proof_OMA) // proof DED
            yield break;

        OMA unequal_circles_OMA = (OMA)proof_OMA.arguments[0]; // unequal

        string circleAUri = ((OMS)unequal_circles_OMA.arguments[0]).uri;
        string circleBUri = ((OMS)unequal_circles_OMA.arguments[1]).uri;

        if (!FactRecorder.AllFacts.ContainsKey(circleAUri)
         || !FactRecorder.AllFacts.ContainsKey(circleBUri))
            yield break;

        ret.Add(new UnEqualCirclesFact(circleAUri, circleBUri, fact.@ref));
    }

    /// \copydoc Fact.generateLabel
    protected override string generateLabel(FactRecorder name_space)
        => Circle1.GetLabel(name_space) + " = " + Circle2.GetLabel(name_space);

    public override MMTFact MakeMMTDeclaration()
    {
        SOMDoc tp = new OMA(
            new OMS(MMTConstants.Ded),
            new[] {
                new OMA(new OMS(MMTConstants.UnEqualityCircles),
                new[]  {
                    new OMS(Cid1),
                    new OMS(Cid2),
        }),});

        return new MMTGeneralFact(_LastLabel, tp, Defines());
    }

    public override SOMDoc Defines()
        => null;

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

    /// \copydoc Fact.getDependentFactIds
    protected override string[] GetDependentFactIds()
        => new string[] { Cid1, Cid2 };

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