nwjh/LLMServe/wired_table_rec/utils_table_line_rec.py

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2025-03-24 09:27:03 +08:00
# -*- encoding: utf-8 -*-
# @Author: SWHL
# @Contact: liekkaskono@163.com
import copy
import math
import cv2
import numpy as np
from scipy.spatial import distance as dist
from skimage import measure
def bbox_decode(heat, wh, reg=None, K=100):
"""bbox组成[V1, V2, V3, V4]
V1~V4: bbox的4个坐标点
"""
batch = heat.shape[0]
heat, keep = _nms(heat)
scores, inds, clses, ys, xs = _topk(heat, K=K)
if reg is not None:
reg = _tranpose_and_gather_feat(reg, inds)
reg = reg.reshape(batch, K, 2)
xs = xs.reshape(batch, K, 1) + reg[:, :, 0:1]
ys = ys.reshape(batch, K, 1) + reg[:, :, 1:2]
else:
xs = xs.reshape(batch, K, 1) + 0.5
ys = ys.reshape(batch, K, 1) + 0.5
wh = _tranpose_and_gather_feat(wh, inds)
wh = wh.reshape(batch, K, 8)
clses = clses.reshape(batch, K, 1).astype(np.float32)
scores = scores.reshape(batch, K, 1)
bboxes = np.concatenate(
[
xs - wh[..., 0:1],
ys - wh[..., 1:2],
xs - wh[..., 2:3],
ys - wh[..., 3:4],
xs - wh[..., 4:5],
ys - wh[..., 5:6],
xs - wh[..., 6:7],
ys - wh[..., 7:8],
],
axis=2,
)
detections = np.concatenate([bboxes, scores, clses], axis=2)
return detections, inds
def _nms(heat, kernel=3):
pad = (kernel - 1) // 2
hmax = max_pool(heat, kernel_size=kernel, stride=1, padding=pad)
keep = hmax == heat
return heat * keep, keep
def max_pool(img, kernel_size, stride, padding):
h, w = img.shape[2:]
img = np.pad(
img,
((0, 0), (0, 0), (padding, padding), (padding, padding)),
"constant",
constant_values=0,
)
res_h = ((h + 2 - kernel_size) // stride) + 1
res_w = ((w + 2 - kernel_size) // stride) + 1
res = np.zeros((img.shape[0], img.shape[1], res_h, res_w))
for i in range(res_h):
for j in range(res_w):
temp = img[
:,
:,
i * stride : i * stride + kernel_size,
j * stride : j * stride + kernel_size,
]
res[:, :, i, j] = temp.max()
return res
def _topk(scores, K=40):
batch, cat, height, width = scores.shape
topk_scores, topk_inds = find_topk(scores.reshape(batch, cat, -1), K)
topk_inds = topk_inds % (height * width)
topk_ys = topk_inds / width
topk_xs = np.float32(np.int32(topk_inds % width))
topk_score, topk_ind = find_topk(topk_scores.reshape(batch, -1), K)
topk_clses = np.int32(topk_ind / K)
topk_inds = _gather_feat(topk_inds.reshape(batch, -1, 1), topk_ind).reshape(
batch, K
)
topk_ys = _gather_feat(topk_ys.reshape(batch, -1, 1), topk_ind).reshape(batch, K)
topk_xs = _gather_feat(topk_xs.reshape(batch, -1, 1), topk_ind).reshape(batch, K)
return topk_score, topk_inds, topk_clses, topk_ys, topk_xs
def find_topk(a, k, axis=-1, largest=True, sorted=True):
if axis is None:
axis_size = a.size
else:
axis_size = a.shape[axis]
assert 1 <= k <= axis_size
a = np.asanyarray(a)
if largest:
index_array = np.argpartition(a, axis_size - k, axis=axis)
topk_indices = np.take(index_array, -np.arange(k) - 1, axis=axis)
else:
index_array = np.argpartition(a, k - 1, axis=axis)
topk_indices = np.take(index_array, np.arange(k), axis=axis)
topk_values = np.take_along_axis(a, topk_indices, axis=axis)
if sorted:
sorted_indices_in_topk = np.argsort(topk_values, axis=axis)
if largest:
sorted_indices_in_topk = np.flip(sorted_indices_in_topk, axis=axis)
sorted_topk_values = np.take_along_axis(
topk_values, sorted_indices_in_topk, axis=axis
)
sorted_topk_indices = np.take_along_axis(
topk_indices, sorted_indices_in_topk, axis=axis
)
return sorted_topk_values, sorted_topk_indices
return topk_values, topk_indices
def _gather_feat(feat, ind):
dim = feat.shape[2]
ind = np.broadcast_to(ind[:, :, None], (ind.shape[0], ind.shape[1], dim))
feat = _gather_np(feat, 1, ind)
return feat
def _gather_np(data, dim, index):
"""
Gathers values along an axis specified by dim.
For a 3-D tensor the output is specified by:
out[i][j][k] = input[index[i][j][k]][j][k] # if dim == 0
out[i][j][k] = input[i][index[i][j][k]][k] # if dim == 1
out[i][j][k] = input[i][j][index[i][j][k]] # if dim == 2
:param dim: The axis along which to index
:param index: A tensor of indices of elements to gather
:return: tensor of gathered values
"""
idx_xsection_shape = index.shape[:dim] + index.shape[dim + 1 :]
data_xsection_shape = data.shape[:dim] + data.shape[dim + 1 :]
if idx_xsection_shape != data_xsection_shape:
raise ValueError(
"Except for dimension "
+ str(dim)
+ ", all dimensions of index and data should be the same size"
)
if index.dtype != np.int64:
raise TypeError("The values of index must be integers")
data_swaped = np.swapaxes(data, 0, dim)
index_swaped = np.swapaxes(index, 0, dim)
gathered = np.take_along_axis(data_swaped, index_swaped, axis=0)
return np.swapaxes(gathered, 0, dim)
def _tranpose_and_gather_feat(feat, ind):
feat = np.ascontiguousarray(np.transpose(feat, [0, 2, 3, 1]))
feat = feat.reshape(feat.shape[0], -1, feat.shape[3])
feat = _gather_feat(feat, ind)
return feat
def gbox_decode(mk, st_reg, reg=None, K=400):
"""gbox的组成[V1, P1, P2, P3, P4]
P1~P4: 四个框的中心点
V1: 四个框的交点
"""
batch = mk.shape[0]
mk, keep = _nms(mk)
scores, inds, clses, ys, xs = _topk(mk, K=K)
if reg is not None:
reg = _tranpose_and_gather_feat(reg, inds)
reg = reg.reshape(batch, K, 2)
xs = xs.reshape(batch, K, 1) + reg[:, :, 0:1]
ys = ys.reshape(batch, K, 1) + reg[:, :, 1:2]
else:
xs = xs.reshape(batch, K, 1) + 0.5
ys = ys.reshape(batch, K, 1) + 0.5
scores = scores.reshape(batch, K, 1)
clses = clses.reshape(batch, K, 1).astype(np.float32)
st_Reg = _tranpose_and_gather_feat(st_reg, inds)
bboxes = np.concatenate(
[
xs - st_Reg[..., 0:1],
ys - st_Reg[..., 1:2],
xs - st_Reg[..., 2:3],
ys - st_Reg[..., 3:4],
xs - st_Reg[..., 4:5],
ys - st_Reg[..., 5:6],
xs - st_Reg[..., 6:7],
ys - st_Reg[..., 7:8],
],
axis=2,
)
return np.concatenate([xs, ys, bboxes, scores, clses], axis=2), keep
def transform_preds(coords, center, scale, output_size, rot=0):
target_coords = np.zeros(coords.shape)
trans = get_affine_transform(center, scale, rot, output_size, inv=1)
for p in range(coords.shape[0]):
target_coords[p, 0:2] = affine_transform(coords[p, 0:2], trans)
return target_coords
def get_affine_transform(
center, scale, rot, output_size, shift=np.array([0, 0], dtype=np.float32), inv=0
):
if not isinstance(scale, np.ndarray) and not isinstance(scale, list):
scale = np.array([scale, scale], dtype=np.float32)
scale_tmp = scale
src_w = scale_tmp[0]
dst_w = output_size[0]
dst_h = output_size[1]
rot_rad = np.pi * rot / 180
src_dir = get_dir([0, src_w * -0.5], rot_rad)
dst_dir = np.array([0, dst_w * -0.5], np.float32)
src = np.zeros((3, 2), dtype=np.float32)
dst = np.zeros((3, 2), dtype=np.float32)
src[0, :] = center + scale_tmp * shift
src[1, :] = center + src_dir + scale_tmp * shift
dst[0, :] = [dst_w * 0.5, dst_h * 0.5]
dst[1, :] = np.array([dst_w * 0.5, dst_h * 0.5], np.float32) + dst_dir
src[2:, :] = get_3rd_point(src[0, :], src[1, :])
dst[2:, :] = get_3rd_point(dst[0, :], dst[1, :])
if inv:
trans = cv2.getAffineTransform(np.float32(dst), np.float32(src))
else:
trans = cv2.getAffineTransform(np.float32(src), np.float32(dst))
return trans
def affine_transform(pt, t):
new_pt = np.array([pt[0], pt[1], 1.0], dtype=np.float32).T
new_pt = np.dot(t, new_pt)
return new_pt[:2]
def get_dir(src_point, rot_rad):
sn, cs = np.sin(rot_rad), np.cos(rot_rad)
src_result = [0, 0]
src_result[0] = src_point[0] * cs - src_point[1] * sn
src_result[1] = src_point[0] * sn + src_point[1] * cs
return src_result
def get_3rd_point(a, b):
direct = a - b
return b + np.array([-direct[1], direct[0]], dtype=np.float32)
def bbox_post_process(bbox, c, s, h, w):
for i in range(bbox.shape[0]):
bbox[i, :, 0:2] = transform_preds(bbox[i, :, 0:2], c[i], s[i], (w, h))
bbox[i, :, 2:4] = transform_preds(bbox[i, :, 2:4], c[i], s[i], (w, h))
bbox[i, :, 4:6] = transform_preds(bbox[i, :, 4:6], c[i], s[i], (w, h))
bbox[i, :, 6:8] = transform_preds(bbox[i, :, 6:8], c[i], s[i], (w, h))
return bbox
def gbox_post_process(gbox, c, s, h, w):
for i in range(gbox.shape[0]):
gbox[i, :, 0:2] = transform_preds(gbox[i, :, 0:2], c[i], s[i], (w, h))
gbox[i, :, 2:4] = transform_preds(gbox[i, :, 2:4], c[i], s[i], (w, h))
gbox[i, :, 4:6] = transform_preds(gbox[i, :, 4:6], c[i], s[i], (w, h))
gbox[i, :, 6:8] = transform_preds(gbox[i, :, 6:8], c[i], s[i], (w, h))
gbox[i, :, 8:10] = transform_preds(gbox[i, :, 8:10], c[i], s[i], (w, h))
return gbox
def nms(dets, thresh):
if len(dets) < 2:
return dets
index_keep, keep = [], []
for i in range(len(dets)):
box = dets[i]
if box[-1] < thresh:
break
max_score_index = -1
ctx = (dets[i][0] + dets[i][2] + dets[i][4] + dets[i][6]) / 4
cty = (dets[i][1] + dets[i][3] + dets[i][5] + dets[i][7]) / 4
for j in range(len(dets)):
if i == j or dets[j][-1] < thresh:
break
x1, y1 = dets[j][0], dets[j][1]
x2, y2 = dets[j][2], dets[j][3]
x3, y3 = dets[j][4], dets[j][5]
x4, y4 = dets[j][6], dets[j][7]
a = (x2 - x1) * (cty - y1) - (y2 - y1) * (ctx - x1)
b = (x3 - x2) * (cty - y2) - (y3 - y2) * (ctx - x2)
c = (x4 - x3) * (cty - y3) - (y4 - y3) * (ctx - x3)
d = (x1 - x4) * (cty - y4) - (y1 - y4) * (ctx - x4)
if all(x > 0 for x in (a, b, c, d)) or all(x < 0 for x in (a, b, c, d)):
if dets[i][8] > dets[j][8] and max_score_index < 0:
max_score_index = i
elif dets[i][8] < dets[j][8]:
max_score_index = -2
break
if max_score_index > -1:
index_keep.append(max_score_index)
elif max_score_index == -1:
index_keep.append(i)
keep = [dets[index_keep[i]] for i in range(len(index_keep))]
return np.array(keep)
def group_bbox_by_gbox(
bboxes, gboxes, score_thred=0.3, v2c_dist_thred=2, c2v_dist_thred=0.5
):
def point_in_box(box, point):
x1, y1, x2, y2 = box[0], box[1], box[2], box[3]
x3, y3, x4, y4 = box[4], box[5], box[6], box[7]
ctx, cty = point[0], point[1]
a = (x2 - x1) * (cty - y1) - (y2 - y1) * (ctx - x1)
b = (x3 - x2) * (cty - y2) - (y3 - y2) * (ctx - x2)
c = (x4 - x3) * (cty - y3) - (y4 - y3) * (ctx - x3)
d = (x1 - x4) * (cty - y4) - (y1 - y4) * (ctx - x4)
if all(x > 0 for x in (a, b, c, d)) or all(x < 0 for x in (a, b, c, d)):
return True
return False
def get_distance(pt1, pt2):
return math.sqrt(
(pt1[0] - pt2[0]) * (pt1[0] - pt2[0])
+ (pt1[1] - pt2[1]) * (pt1[1] - pt2[1])
)
dets = copy.deepcopy(bboxes)
sign = np.zeros((len(dets), 4))
for gbox in gboxes:
if gbox[10] < score_thred:
break
vertex = [gbox[0], gbox[1]]
for i in range(4):
center = [gbox[2 * i + 2], gbox[2 * i + 3]]
if get_distance(vertex, center) < v2c_dist_thred:
continue
for k, bbox in enumerate(dets):
if bbox[8] < score_thred:
break
if sum(sign[k]) == 4:
continue
w = (abs(bbox[6] - bbox[0]) + abs(bbox[4] - bbox[2])) / 2
h = (abs(bbox[3] - bbox[1]) + abs(bbox[5] - bbox[7])) / 2
m = max(w, h)
if point_in_box(bbox, center):
min_dist, min_id = 1e4, -1
for j in range(4):
dist = get_distance(vertex, [bbox[2 * j], bbox[2 * j + 1]])
if dist < min_dist:
min_dist = dist
min_id = j
if (
min_id > -1
and min_dist < c2v_dist_thred * m
and sign[k][min_id] == 0
):
bboxes[k][2 * min_id] = vertex[0]
bboxes[k][2 * min_id + 1] = vertex[1]
sign[k][min_id] = 1
return bboxes
def get_table_line(binimg, axis=0, lineW=10):
##获取表格线
##axis=0 横线
##axis=1 竖线
labels = measure.label(binimg > 0, connectivity=2) # 8连通区域标记
regions = measure.regionprops(labels)
if axis == 1:
lineboxes = [
min_area_rect(line.coords)
for line in regions
if line.bbox[2] - line.bbox[0] > lineW
]
else:
lineboxes = [
min_area_rect(line.coords)
for line in regions
if line.bbox[3] - line.bbox[1] > lineW
]
return lineboxes
def min_area_rect(coords):
"""
多边形外接矩形
"""
rect = cv2.minAreaRect(coords[:, ::-1])
box = cv2.boxPoints(rect)
box = box.reshape((8,)).tolist()
box = image_location_sort_box(box)
x1, y1, x2, y2, x3, y3, x4, y4 = box
degree, w, h, cx, cy = calculate_center_rotate_angle(box)
if w < h:
xmin = (x1 + x2) / 2
xmax = (x3 + x4) / 2
ymin = (y1 + y2) / 2
ymax = (y3 + y4) / 2
else:
xmin = (x1 + x4) / 2
xmax = (x2 + x3) / 2
ymin = (y1 + y4) / 2
ymax = (y2 + y3) / 2
# degree,w,h,cx,cy = solve(box)
# x1,y1,x2,y2,x3,y3,x4,y4 = box
# return {'degree':degree,'w':w,'h':h,'cx':cx,'cy':cy}
return [xmin, ymin, xmax, ymax]
def image_location_sort_box(box):
x1, y1, x2, y2, x3, y3, x4, y4 = box[:8]
pts = (x1, y1), (x2, y2), (x3, y3), (x4, y4)
pts = np.array(pts, dtype="float32")
(x1, y1), (x2, y2), (x3, y3), (x4, y4) = _order_points(pts)
return [x1, y1, x2, y2, x3, y3, x4, y4]
def calculate_center_rotate_angle(box):
"""
cx,cy点 w,h 旋转 angle 的坐标,能一定程度缓解图片的内部倾斜但是还是依赖模型稳妥
x = cx-w/2
y = cy-h/2
x1-cx = -w/2*cos(angle) +h/2*sin(angle)
y1 -cy= -w/2*sin(angle) -h/2*cos(angle)
h(x1-cx) = -wh/2*cos(angle) +hh/2*sin(angle)
w(y1 -cy)= -ww/2*sin(angle) -hw/2*cos(angle)
(hh+ww)/2sin(angle) = h(x1-cx)-w(y1 -cy)
"""
x1, y1, x2, y2, x3, y3, x4, y4 = box[:8]
cx = (x1 + x3 + x2 + x4) / 4.0
cy = (y1 + y3 + y4 + y2) / 4.0
w = (
np.sqrt((x2 - x1) ** 2 + (y2 - y1) ** 2)
+ np.sqrt((x3 - x4) ** 2 + (y3 - y4) ** 2)
) / 2
h = (
np.sqrt((x2 - x3) ** 2 + (y2 - y3) ** 2)
+ np.sqrt((x1 - x4) ** 2 + (y1 - y4) ** 2)
) / 2
# x = cx-w/2
# y = cy-h/2
sinA = (h * (x1 - cx) - w * (y1 - cy)) * 1.0 / (h * h + w * w) * 2
angle = np.arcsin(sinA)
return angle, w, h, cx, cy
def _order_points(pts):
# 根据x坐标对点进行排序
"""
---------------------
本项目中是为了排序后得到[(xmin,ymin),(xmax,ymin),(xmax,ymax),(xmin,ymax)]
作者Tong_T
来源CSDN
原文https://blog.csdn.net/Tong_T/article/details/81907132
版权声明本文为博主原创文章转载请附上博文链接
"""
x_sorted = pts[np.argsort(pts[:, 0]), :]
left_most = x_sorted[:2, :]
right_most = x_sorted[2:, :]
left_most = left_most[np.argsort(left_most[:, 1]), :]
(tl, bl) = left_most
distance = dist.cdist(tl[np.newaxis], right_most, "euclidean")[0]
(br, tr) = right_most[np.argsort(distance)[::-1], :]
return np.array([tl, tr, br, bl], dtype="float32")
def sqrt(p1, p2):
return np.sqrt((p1[0] - p2[0]) ** 2 + (p1[1] - p2[1]) ** 2)
def adjust_lines(lines, alph=50, angle=50):
lines_n = len(lines)
new_lines = []
for i in range(lines_n):
x1, y1, x2, y2 = lines[i]
cx1, cy1 = (x1 + x2) / 2, (y1 + y2) / 2
for j in range(lines_n):
if i != j:
x3, y3, x4, y4 = lines[j]
cx2, cy2 = (x3 + x4) / 2, (y3 + y4) / 2
if (x3 < cx1 < x4 or y3 < cy1 < y4) or (
x1 < cx2 < x2 or y1 < cy2 < y2
): # 判断两个横线在y方向的投影重不重合
continue
else:
r = sqrt((x1, y1), (x3, y3))
k = abs((y3 - y1) / (x3 - x1 + 1e-10))
a = math.atan(k) * 180 / math.pi
if r < alph and a < angle:
new_lines.append((x1, y1, x3, y3))
r = sqrt((x1, y1), (x4, y4))
k = abs((y4 - y1) / (x4 - x1 + 1e-10))
a = math.atan(k) * 180 / math.pi
if r < alph and a < angle:
new_lines.append((x1, y1, x4, y4))
r = sqrt((x2, y2), (x3, y3))
k = abs((y3 - y2) / (x3 - x2 + 1e-10))
a = math.atan(k) * 180 / math.pi
if r < alph and a < angle:
new_lines.append((x2, y2, x3, y3))
r = sqrt((x2, y2), (x4, y4))
k = abs((y4 - y2) / (x4 - x2 + 1e-10))
a = math.atan(k) * 180 / math.pi
if r < alph and a < angle:
new_lines.append((x2, y2, x4, y4))
return new_lines
def final_adjust_lines(rowboxes, colboxes):
nrow = len(rowboxes)
ncol = len(colboxes)
for i in range(nrow):
for j in range(ncol):
rowboxes[i] = line_to_line(rowboxes[i], colboxes[j], alpha=20, angle=30)
colboxes[j] = line_to_line(colboxes[j], rowboxes[i], alpha=20, angle=30)
return rowboxes, colboxes
def draw_lines(im, bboxes, color=(0, 0, 0), lineW=3):
"""
boxes: bounding boxes
"""
tmp = np.copy(im)
c = color
h, w = im.shape[:2]
for box in bboxes:
x1, y1, x2, y2 = box[:4]
cv2.line(
tmp, (int(x1), int(y1)), (int(x2), int(y2)), c, lineW, lineType=cv2.LINE_AA
)
return tmp
def line_to_line(points1, points2, alpha=10, angle=30):
"""
线段之间的距离
"""
x1, y1, x2, y2 = points1
ox1, oy1, ox2, oy2 = points2
xy = np.array([(x1, y1), (x2, y2)], dtype="float32")
A1, B1, C1 = fit_line(xy)
oxy = np.array([(ox1, oy1), (ox2, oy2)], dtype="float32")
A2, B2, C2 = fit_line(oxy)
flag1 = point_line_cor(np.array([x1, y1], dtype="float32"), A2, B2, C2)
flag2 = point_line_cor(np.array([x2, y2], dtype="float32"), A2, B2, C2)
if (flag1 > 0 and flag2 > 0) or (flag1 < 0 and flag2 < 0): # 横线或者竖线在竖线或者横线的同一侧
if (A1 * B2 - A2 * B1) != 0:
x = (B1 * C2 - B2 * C1) / (A1 * B2 - A2 * B1)
y = (A2 * C1 - A1 * C2) / (A1 * B2 - A2 * B1)
# x, y = round(x, 2), round(y, 2)
p = (x, y) # 横线与竖线的交点
r0 = sqrt(p, (x1, y1))
r1 = sqrt(p, (x2, y2))
if min(r0, r1) < alpha: # 若交点与线起点或者终点的距离小于alpha则延长线到交点
if r0 < r1:
k = abs((y2 - p[1]) / (x2 - p[0] + 1e-10))
a = math.atan(k) * 180 / math.pi
if a < angle or abs(90 - a) < angle:
points1 = np.array([p[0], p[1], x2, y2], dtype="float32")
else:
k = abs((y1 - p[1]) / (x1 - p[0] + 1e-10))
a = math.atan(k) * 180 / math.pi
if a < angle or abs(90 - a) < angle:
points1 = np.array([x1, y1, p[0], p[1]], dtype="float32")
return points1
def min_area_rect_box(
regions, flag=True, W=0, H=0, filtersmall=False, adjust_box=False
):
"""
多边形外接矩形
"""
boxes = []
for region in regions:
if region.bbox_area > H * W * 3 / 4: # 过滤大的单元格
continue
rect = cv2.minAreaRect(region.coords[:, ::-1])
box = cv2.boxPoints(rect)
box = box.reshape((8,)).tolist()
box = image_location_sort_box(box)
x1, y1, x2, y2, x3, y3, x4, y4 = box
angle, w, h, cx, cy = calculate_center_rotate_angle(box)
# if adjustBox:
# x1, y1, x2, y2, x3, y3, x4, y4 = xy_rotate_box(cx, cy, w + 5, h + 5, angle=0, degree=None)
# x1, x4 = max(x1, 0), max(x4, 0)
# y1, y2 = max(y1, 0), max(y2, 0)
# if w > 32 and h > 32 and flag:
# if abs(angle / np.pi * 180) < 20:
# if filtersmall and (w < 10 or h < 10):
# continue
# boxes.append([x1, y1, x2, y2, x3, y3, x4, y4])
# else:
if w * h < 0.5 * W * H:
if filtersmall and (
w < 15 or h < 15
): # or w / h > 30 or h / w > 30): # 过滤小的单元格
continue
boxes.append([x1, y1, x2, y2, x3, y3, x4, y4])
return boxes
def point_line_cor(p, A, B, C):
##判断点与线之间的位置关系
# 一般式直线方程(Ax+By+c)=0
x, y = p
r = A * x + B * y + C
return r
def fit_line(p):
"""A = Y2 - Y1
B = X1 - X2
C = X2*Y1 - X1*Y2
AX+BY+C=0
直线一般方程
"""
x1, y1 = p[0]
x2, y2 = p[1]
A = y2 - y1
B = x1 - x2
C = x2 * y1 - x1 * y2
return A, B, C