Wrong outcome for mathematically correct circle segment 180 degrees

Question

I am trying to program an add-on for Blender that would make a perfect circle segment.

Pure mathematics. No bmesh or bpy.

The green vertices are my result and the blue circle is what I am after

With h1 = (h + (sqrt(a**2 - (math.cos(angle)*a)**2) - a)) you can calculate the height offset of all vertices, but the operator menu will not allow you to change the height anymore.

So verts1.append((math.cos(angle)*a, math.sin(angle)*h, 0.0)) will allow you to change height and diameter but the outcome is wrong. As sees on the screenshot above.

While verts1.append((math.cos(angle)*a, h1, 0.0)) moves all vertices up or down as you can see in this screenshot below.

My goal is to keep the lowest vertices on the X axis as in the first screenshot.

The circle is radius 1000mm (1M) and moved -500 Y. The height of the segment is 500 and diameter is 1732mm

This is the code so far. Upon running a new "button" will appear under add --> mesh --> Segment circle

bl_info = {
    "name": "segment",
    "author": "The Bold Guy",
    "version": (1, 0),
    "blender": (2, 80, 1),
    "location": "View3D > Add > Mesh > segment",
    "description": "Adds a new Mesh Object",
    "warning": "",
    "wiki_url": "",
    "category": "Add Mesh",
}


import bpy
import bmesh
import math
import mathutils
from mathutils import Matrix
from bpy import types
from bpy.props import (
    StringProperty,
    BoolProperty,
    IntProperty,
    IntVectorProperty,
    FloatProperty,
    FloatVectorProperty,
    EnumProperty,
    PointerProperty,
)
from bpy.types import (
    Panel,
    Menu,
    Operator,
    PropertyGroup,
)
from bpy_extras.object_utils import AddObjectHelper, object_data_add
from mathutils import Vector
from math import degrees, radians, pi, tan, asin, sin, cos, acos, sqrt



def add_plane_mesh(self, context):

    mesh = bpy.data.meshes.new("Vert")

    verts = []
    edges = []
    faces = []


    h = self.height
    a = self.diam/2
    h1 = []
   


    for i in range(self.segments+1):
        angle = (math.pi * (i / self.segments))
       

        ##Calculate the offset of vertices on any given point of circle (180 degrees)
        #(math.cos(angle)*a)
 
        ##Calculating Height of an Arc at Any Point
        h1 = (h + (sqrt(a**2 - (math.cos(angle)*a)**2) - a))
        
        
        ##make list X,Y,Z for vertices to be added
        verts.append((math.cos(angle)*a, math.sin(angle)*h, 0.0))
 


    mesh.from_pydata(verts, edges, faces)   

    return mesh


def add_plane_mesh1(self, context):


    return {"FINISHED"}


class OBJECT_Oval_pipe_add_object(Operator, AddObjectHelper):
    """Create a new Mesh Object"""

    bl_idname = "tools.add_segment"
    bl_label = "Add segment"
    bl_options = {"REGISTER", "UNDO", 'PRESET'}

    segments: IntProperty(
        name="Segments",
        # unit='LENGTH',
        min=1,
        default=8,
    )
    angle: IntProperty(
        name="Segments",
        # unit='LENGTH',
        min=0,
        max=360,
        default=180,
    )
    height: FloatProperty(
        attr='height', name="Height", unit="LENGTH", default=1000 / 1000 ,
    )
    diam: FloatProperty(
        attr='diam', name="length", unit="LENGTH", default=2000 / 1000,
    )



    def execute(self, context):


        object_data_add(context, add_plane_mesh(self, context), operator=self)
        return {"FINISHED"}


# Registration
def add_object_button(self, context):
    self.layout.operator(
        OBJECT_Oval_pipe_add_object.bl_idname, text="Segment Circle", icon="PLUGIN"
    )


# This allows you to right click on a button and link to documentation
def add_object_manual_map():
    url_manual_prefix = "https://docs.blender.org/manual/en/latest/"
    url_manual_mapping = (
        ("bpy.ops.mesh.add_object", "scene_layout/object/types.html"),
    )
    return url_manual_prefix, url_manual_mapping


def register():
    bpy.utils.register_class(OBJECT_Oval_pipe_add_object)
    bpy.utils.register_manual_map(add_object_manual_map)
    bpy.types.VIEW3D_MT_mesh_add.append(add_object_button)


def unregister():
    bpy.utils.unregister_class(OBJECT_Oval_pipe_add_object)
    bpy.utils.unregister_manual_map(add_object_manual_map)
    bpy.types.VIEW3D_MT_mesh_add.remove(add_object_button)


if __name__ == "__main__":
    register()

Answer 1

IIUC the question is how to draw the part of the circle $(x, y) = (r \cos \theta, r \sin \theta - h)$ that is above the x-axis. In that case, like Ron Jensen says, you just need to calculate the angle to start and end at.

    h = self.height
    diam = abs(self.diam)
    r = diam/2

    if r == 0:
        # No circle
        return mesh
    elif h >= r:
        # Circle entirely below x-axis 
        return mesh
    elif h <= -r:
        # Circle entirely above x-axis
        angle_start = -math.pi/2
        angle_end = math.pi*3/2
    else:
        # Circle partially above x-axis
        angle_start = math.asin(h/r)
        angle_end = math.pi - angle_start

    for i in range(self.segments+1):
        factor = i / self.segments
        angle = angle_start*(1 - factor) + angle_end*factor
        verts.append((math.cos(angle)*r, math.sin(angle)*r - h, 0.0))