diff --git a/data/topo_TIN.log b/data/topo_TIN.log
index 83494e8..9b0b55d 100644
--- a/data/topo_TIN.log
+++ b/data/topo_TIN.log
@@ -331,1919 +331,1561 @@
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diff --git a/data/topo_TIN.msh b/data/topo_TIN.msh
index 49790c2..8351de2 100644
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-1917 530 790 18.2201
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-1932 80 680 40.8898
-1933 60 680 42.7726
-1934 90 670 42.061
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-1937 580 840 10.9627
-1938 560 860 11.5632
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-1940 310 660 107.36
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-1943 500 940 9.2818
-1944 180 200 13.6473
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-1947 790 680 15.1529
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-1952 920 810 12.1773
-1953 400 450 150.006
-1954 490 530 117.233
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-1956 400 120 19.3401
-1957 600 460 157.333
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-1959 340 380 46.2026
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-1963 540 230 47.921
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-1966 770 560 36.8935
-1967 280 80 12.7139
-1968 300 60 12.2695
-1969 940 460 28.2321
-1970 740 120 22.5295
-1971 960 240 15.1082
-1972 980 260 15.6046
-1973 840 540 28.2358
-1974 490 560 98.3386
-1975 330 350 34.5849
-1976 340 750 86.67659999999999
-1977 350 750 84.5986
-1978 330 750 89.6465
-1979 240 570 45.0428
-1980 520 470 168.821
-1981 510 470 168.277
-1982 700 420 115.744
-1983 260 800 53.4204
-1984 910 280 26.9883
-1985 900 260 26.7625
-1986 800 300 61.84
-1987 650 440 149.841
-1988 360 240 23.0771
-1989 350 220 22.0695
-1990 370 200 21.7845
-1991 770 230 50.4861
-1992 790 490 47.2694
-1993 780 480 52.7377
-1994 980 460 24.236
-1995 680 280 95.7726
-1996 670 280 100.314
-1997 760 600 30.4959
-1998 760 620 25.9011
-1999 380 740 73.00069999999999
-2000 620 260 81.3356
-2001 720 240 59.0792
-2002 420 440 161.079
-2003 420 450 166.249
-2004 370 400 69.9794
-2005 390 510 149.568
-2006 910 140 18.9188
-2007 720 330 99.4134
-2008 500 760 32.381
-2009 520 200 37.4038
-2010 500 200 39.2218
-2011 530 180 34.8147
-2012 90 760 42.899
-2013 610 280 108.375
-2014 120 300 14.7709
-2015 610 380 193.159
-2016 330 740 93.21169999999999
-2017 920 240 23.8297
-2018 920 260 22.6542
-2019 350 360 43.2286
-2020 780 390 56.9199
-2021 530 580 89.59999999999999
-2022 730 390 93.4409
-2023 370 410 78.7077
-2024 430 490 164.523
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diff --git a/data/topo_TIN.neigh b/data/topo_TIN.neigh
index 77e6324..3886354 100644
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diff --git a/src/demo/demo.cpp b/src/demo/demo.cpp
index 2c15085..65eb108 100644
--- a/src/demo/demo.cpp
+++ b/src/demo/demo.cpp
@@ -14,10 +14,20 @@ int main(int argc, char const *argv[])
 	}
 	infile.close();
 
+	std::vector<vertex2dc> box(4);
+	box[0].id = 0; box[0].x = 500; box[0].y = 250;
+	box[1].id = 1; box[1].x = 750; box[1].y = 250;
+	box[2].id = 2; box[2].x = 750; box[2].y = 500;
+	box[3].id = 3; box[3].x = 500; box[3].y = 500;
+
+	std::vector<region> box_region(1);
+	box_region[0].set(box, 5.0);
+
 	std::vector<double> err_records;
 	std::vector<vertex2dc*> tin_vert;
 	std::vector<triangle*> tin_ele;
-	grd2tin(topo, 0, 1000, 0, 1000, 10, 10, tin_vert, tin_ele, 1.0, nullptr, &err_records);
+	//grd2tin(topo, 0, 1000, 0, 1000, 10, 10, tin_vert, tin_ele, 1.0, nullptr, &err_records, nullptr);
+	grd2tin(topo, 0, 1000, 0, 1000, 10, 10, tin_vert, tin_ele, 1.0, nullptr, &err_records, &box_region);
 
 	// Write a log file
 	std::ofstream logfile("data/topo_TIN.log");
diff --git a/src/lib/tin.cpp b/src/lib/tin.cpp
index 92a7d89..a153976 100644
--- a/src/lib/tin.cpp
+++ b/src/lib/tin.cpp
@@ -13,6 +13,10 @@
  */
 
 #include "tin.h"
+#include "cmath"
+#include "algorithm"
+#include "exception"
+#include "stdexcept"
 
 // Start vertex definition
 vertex2dc::vertex2dc()
@@ -21,12 +25,12 @@ vertex2dc::vertex2dc()
     id = 0;
 }
 
-vertex2dc::vertex2dc(double inx, double iny, double inelev, unsigned int inid)
+vertex2dc::vertex2dc(double inx, double iny, double inelev, size_t inid)
 {
     set(inx, iny, inelev, inid);
 }
 
-void vertex2dc::set(double inx, double iny, double inelev, unsigned int inid)
+void vertex2dc::set(double inx, double iny, double inelev, size_t inid)
 {
     x = inx; y = iny; elev = inelev; id = inid;
     return;
@@ -42,6 +46,15 @@ bool operator ==(const vertex2dc &a, const vertex2dc &b)
 	return false;
 }
 
+bool duplicated_vertex(const vertex2dc &a, const vertex2dc &b)
+{
+	if (a == b && a.id == b.id)
+	{
+		return true;
+	}
+	return false;
+}
+
 // Test if the three points are on the same line
 bool is_collinear(vertex2dc *a_ptr, vertex2dc *b_ptr, vertex2dc *c_ptr)
 {
@@ -62,16 +75,47 @@ void circumcircle(vertex2dc *v0, vertex2dc *v1, vertex2dc *v2, double &cx, doubl
 
 	cx = ((v2->y - v0->y)*m + (v0->y - v1->y)*u)*s;
 	cy = ((v0->x - v2->x)*m + (v1->x - v0->x)*u)*s;
-	cr = (v0->x - cx)*(v0->x - cx) + (v0->y - cy)*(v0->y - cy); // not need to calculate the squared root here
+	// not need to calculate the squared root here
+	cr = (v0->x - cx)*(v0->x - cx) + (v0->y - cy)*(v0->y - cy);
 	return;
 }
 // End vertex definition
 
+// Start region definition
+region::region()
+{
+	maxi_err = 0.0;
+}
+
+region::region(const std::vector<vertex2dc> &outline_vert, double err)
+{
+	set(outline_vert, err);
+}
+
+void region::set(const std::vector<vertex2dc> &outline_vert, double err)
+{
+	if (err <= 0.0)
+	{
+		throw std::invalid_argument("Invalid maximal allowed error.");
+	}
+
+	if (outline_vert.size() < 3)
+	{
+		throw std::runtime_error("The input polygon must have at least 3 vertexes.");
+	}
+	
+	maxi_err = err;
+	poly_vert = outline_vert;
+	return;
+}
+// End region definition
+
 // Start DEM definition
 dem_point::dem_point()
 {
     x = y = elev = NAN;
     err = 0.0;
+	allowed_err = -1.0;
     host = nullptr;
 }
 
@@ -82,11 +126,14 @@ dem_point::dem_point(double inx, double iny, double inelev)
 
 void dem_point::set(double inx, double iny, double inelev)
 {
-    x = inx; y = iny; elev = inelev; err = 0.0; host = nullptr;
+    x = inx; y = iny; elev = inelev;
+	err = 0.0;
+	allowed_err = -1.0;
+	host = nullptr;
     return;
 }
 
-bool compare_dem_point(dem_point *a, dem_point *b) // determination function for std::sort
+bool compare_dem_error(dem_point *a, dem_point *b) // determination function for std::sort
 {
 	if (a->err > b->err) return true;
 	return false;
@@ -102,13 +149,13 @@ triangle::triangle()
 
 triangle::triangle(vertex2dc *v0ptr, vertex2dc *v1ptr, vertex2dc *v2ptr)
 {
-    set(v0ptr, v1ptr, v2ptr);
+    set_vertex(v0ptr, v1ptr, v2ptr);
+	neigh[0] = neigh[1] = neigh[2] = nullptr;
 }
 
-void triangle::set(vertex2dc *v0ptr, vertex2dc *v1ptr, vertex2dc *v2ptr)
+void triangle::set_vertex(vertex2dc *v0ptr, vertex2dc *v1ptr, vertex2dc *v2ptr)
 {
     vert[0] = v0ptr; vert[1] = v1ptr; vert[2] = v2ptr;
-    neigh[0] = neigh[1] = neigh[2] = nullptr;
     circumcircle(vert[0], vert[1], vert[2], cx, cy, cr);
     return;
 }
@@ -256,17 +303,27 @@ void flip_neighboring_triangles(triangle *t, triangle *n, int t_id, int n_id)
 	{
 		tmp_dem = n->hosted_dem[i];
 		tmp_dem->err = fabs(n->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
+		// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+		if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+		{
+			tmp_dem->err = 0.0;
+		}
 	}
 
 	for (int i = 0; i < t->hosted_dem.size(); i++)
 	{
 		tmp_dem = t->hosted_dem[i];
 		tmp_dem->err = fabs(t->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
+		// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+		if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+		{
+			tmp_dem->err = 0.0;
+		}
 	}
 
 	// Sort maximal errors for triangles t and n
-	std::sort(t->hosted_dem.begin(), t->hosted_dem.end(), compare_dem_point);
-	std::sort(n->hosted_dem.begin(), n->hosted_dem.end(), compare_dem_point);
+	std::sort(t->hosted_dem.begin(), t->hosted_dem.end(), compare_dem_error);
+	std::sort(n->hosted_dem.begin(), n->hosted_dem.end(), compare_dem_error);
 	return;
 }
 
@@ -396,7 +453,7 @@ triangle *split_triangle(vertex2dc *v, triangle *t, triangle *new_t[4])
 					break;
 				}
 			}
-			return t->neigh[i]; // Return the neighboring tiangle to be deleted
+			return t->neigh[i]; // Return the neighboring triangle to be deleted
 		}
 	}
 
@@ -444,7 +501,7 @@ bool triangle_inside_polygon(triangle *tri_p, std::vector<vertex2dc> *poly_vert)
 	// If any vertexes of the input triangle is outside of the polygon, return false. Otherwise, return true
 	if (poly_vert->size() < 3)
 	{
-		return true;
+		throw std::runtime_error("The input polygon must have at least 3 vertexes.");
 	}
 	else
 	{
@@ -500,7 +557,7 @@ bool triangle_inside_polygon(triangle *tri_p, std::vector<vertex2dc> *poly_vert)
 }
 
 /**
- * @brief      Test if the input locaiton is inside of the given polygon
+ * @brief      Test if the input location is inside of the given polygon
  * 
  * @param x           Input x coordinate
  * @param y           Input y coordinate
@@ -513,7 +570,7 @@ bool location_inside_polygon(double x, double y, std::vector<vertex2dc> *poly_ve
 	// If any vertexes of the input triangle is outside of the polygon, return false. Otherwise, return true
 	if (poly_vert->size() < 3)
 	{
-		return true;
+		throw std::runtime_error("The input polygon must have at least 3 vertexes.");
 	}
 	else
 	{
@@ -566,7 +623,7 @@ bool location_inside_polygon(double x, double y, std::vector<vertex2dc> *poly_ve
 /**
  * @brief      Generate the TIN from the DEM grid
  *
- * @param[in]  dem          Input DEM grid (Ordered from lower left corner to the upper right corner)
+ * @param[in]  dem          Input DEM grid (Ordered from lower left corner to the upper right corner in a row-major fashion)
  * @param[in]  xmin         The minimal coordinate of the DEM grid on the x-axis
  * @param[in]  xmax         The maximal coordinate of the DEM grid on the x-axis
  * @param[in]  ymin         The minimal coordinate of the DEM grid on the y-axis
@@ -580,24 +637,48 @@ bool location_inside_polygon(double x, double y, std::vector<vertex2dc> *poly_ve
  */
 void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ymin, double ymax, double dx, 
 	double dy, std::vector<vertex2dc*> &out_verts, std::vector<triangle*> &out_tris, double maxi_err, 
-	std::vector<vertex2dc> *outline_poly, std::vector<double> *err_records)
+	std::vector<vertex2dc> *outline_poly, std::vector<double> *err_records, 
+	std::vector<region> *regions, std::vector<double> *dem_error)
 {
 	if (!out_verts.empty()) out_verts.clear();
 	if (!out_tris.empty())  out_tris.clear();
 	if (err_records != nullptr && !err_records->empty()) err_records->clear();
 
-	if (dx <= 0.0 || dy <= 0.0 || maxi_err <= 0.0) return;
-	if (xmin >= xmax || ymin >= ymax || (xmin + dx) > xmax || (ymin + dy) > ymax) return;
+	if (maxi_err <= 0.0)
+	{
+		throw std::invalid_argument("Invalid maximal allowed error.");
+	}
+
+	if (dx <= 0.0 || dy <= 0.0 || xmin >= xmax || ymin >= ymax || (xmin + dx) > xmax || (ymin + dy) > ymax)
+	{
+		throw std::invalid_argument("Invalid DEM parameters.");
+	}
 
 	int xnum = round((xmax - xmin)/dx) + 1;
 	int ynum = round((ymax - ymin)/dy) + 1;
 
-	if (dem.size() != xnum*ynum) return;
+	if (dem.size() != xnum*ynum)
+	{
+		throw std::invalid_argument("Invalid size of the input DEM values w.r.t. the DEM parameters.");
+	}
 
-	// Prepare the DEM points
-	dem_point *tmp_dem = nullptr;
-	dem_point *maxi_err_dem = nullptr;
-	std::vector<dem_point*>::iterator d_iter;
+	if (dem_error != nullptr)
+	{
+		if (dem_error->size() != xnum*ynum)
+		{
+			throw std::invalid_argument("Invalid size of the input DEM errors w.r.t. the DEM parameters.");
+		}
+
+		for (size_t i = 0; i < xnum*ynum; i++)
+		{
+			if (dem_error->at(i) <= 0.0)
+			{
+				throw std::runtime_error("Invalid input DEM error value.");		
+			}
+		}
+	}
+
+	// Prepare the initial vertexes and triangles from corner DEM points
 
 	vertex2dc *tmp_vert = nullptr;
 	tmp_vert = new vertex2dc(xmin, ymin, dem[0], out_verts.size()); // lower left corner
@@ -612,9 +693,7 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 	tmp_vert = new vertex2dc(xmin, ymax, dem[xnum*(ynum-1)], out_verts.size()); // upper left corner
 	out_verts.push_back(tmp_vert);
 
-	triangle *old_tri = nullptr, *tmp_tri = nullptr;
-	triangle *cnst_tri[4];
-	std::vector<triangle*>::iterator t_iter;
+	triangle *tmp_tri = nullptr;
 
 	if (!is_collinear(out_verts[0], out_verts[1], out_verts[2])) // Do not create triangle if the vertexes are collinear
 	{
@@ -628,13 +707,18 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 		out_tris.push_back(tmp_tri); tmp_tri = nullptr;
 	}
 
-	if (out_tris.size() != 2) return;
+	if (out_tris.size() != 2)
+	{
+		throw std::runtime_error("Fail to construct the initial triangles.");
+	}
 
 	out_tris[0]->set_neighbor(nullptr, nullptr, out_tris[1]);
 	out_tris[1]->set_neighbor(out_tris[0], nullptr, nullptr);
 
 	// Find host triangle for all DEM locations
+
 	int tmp_id;
+	dem_point *tmp_dem = nullptr;
 	if (outline_poly == nullptr)
 	{
 		for (int i = 0; i < ynum; ++i)
@@ -642,9 +726,11 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 			for (int j = 0; j < xnum; ++j)
 			{
 				tmp_id = j + i*xnum;
-				if (tmp_id != 0 && tmp_id != (xnum-1) && tmp_id != (xnum*ynum-1) && tmp_id != (xnum*(ynum-1))) // the four corners are already used
+				if (tmp_id != 0 && tmp_id != (xnum-1) && 
+					tmp_id != (xnum*ynum-1) && tmp_id != (xnum*(ynum-1))) // the four corners are already used
 				{
 					tmp_dem = new dem_point(xmin + dx*j, ymin + dy*i, dem[j + i*xnum]);
+					
 					for (int t = 0; t < out_tris.size(); ++t)
 					{
 						if (out_tris[t]->bound_location(tmp_dem->x, tmp_dem->y))
@@ -652,6 +738,29 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 							tmp_dem->host = out_tris[t];
 							tmp_dem->err = fabs(out_tris[t]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 							out_tris[t]->hosted_dem.push_back(tmp_dem);
+							// Find region
+							if (regions != nullptr)
+							{
+								for (size_t r = 0; r < regions->size(); r++)
+								{
+									if (location_inside_polygon(tmp_dem->x, tmp_dem->y, &(regions->at(r).poly_vert)))
+									{
+										tmp_dem->allowed_err = regions->at(r).maxi_err;
+										break;
+									}
+								}
+							}
+
+							if (dem_error != nullptr)
+							{
+								tmp_dem->allowed_err = dem_error->at(tmp_id);
+							}
+
+							// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+							if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+							{
+								tmp_dem->err = 0.0;
+							}
 							break; // already found, no need to search more
 						}
 					}
@@ -666,11 +775,14 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 			for (int j = 0; j < xnum; ++j)
 			{
 				tmp_id = j + i*xnum;
+				// the four corners are already used
+				// We also check to see if the candidate dem point is inside the outline polygon
 				if (tmp_id != 0 && tmp_id != (xnum-1) && 
 					tmp_id != (xnum*ynum-1) && tmp_id != (xnum*(ynum-1)) && 
-					location_inside_polygon(xmin+dx*j, ymin+dy*i, outline_poly)) // the four corners are already used
+					location_inside_polygon(xmin+dx*j, ymin+dy*i, outline_poly))
 				{
 					tmp_dem = new dem_point(xmin + dx*j, ymin + dy*i, dem[j + i*xnum]);
+
 					for (int t = 0; t < out_tris.size(); ++t)
 					{
 						if (out_tris[t]->bound_location(tmp_dem->x, tmp_dem->y))
@@ -678,6 +790,29 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 							tmp_dem->host = out_tris[t];
 							tmp_dem->err = fabs(out_tris[t]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 							out_tris[t]->hosted_dem.push_back(tmp_dem);
+							// Find region
+							if (regions != nullptr)
+							{
+								for (size_t r = 0; r < regions->size(); r++)
+								{
+									if (location_inside_polygon(tmp_dem->x, tmp_dem->y, &(regions->at(r).poly_vert)))
+									{
+										tmp_dem->allowed_err = regions->at(r).maxi_err;
+										break;
+									}
+								}
+							}
+
+							if (dem_error != nullptr)
+							{
+								tmp_dem->allowed_err = dem_error->at(tmp_id);
+							}
+
+							// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+							if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+							{
+								tmp_dem->err = 0.0;
+							}
 							break; // already found, no need to search more
 						}
 					}
@@ -688,9 +823,10 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 
 	// Sort hosted_dem in the desceding order with respect to the error. Get pointer of the dem_point with maximal error
 	double maxi_err_tmp = -1.0;
+	dem_point *maxi_err_dem = nullptr;
 	for (int t = 0; t < out_tris.size(); ++t)
 	{
-		std::sort(out_tris[t]->hosted_dem.begin(), out_tris[t]->hosted_dem.end(), compare_dem_point);
+		std::sort(out_tris[t]->hosted_dem.begin(), out_tris[t]->hosted_dem.end(), compare_dem_error);
 		if (out_tris[t]->hosted_dem[0]->err > maxi_err_tmp)
 		{
 			maxi_err_tmp = out_tris[t]->hosted_dem[0]->err; 
@@ -698,6 +834,9 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 		}
 	}
 
+	triangle *old_tri = nullptr, *cnst_tri[4];
+	std::vector<dem_point*>::iterator d_iter;
+	std::vector<triangle*>::iterator t_iter;
 	while (maxi_err_dem->err >= maxi_err) // quit til the threshold is meet
 	{
 		if (err_records != nullptr)
@@ -768,6 +907,11 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 						tmp_dem->host = cnst_tri[n];
 						tmp_dem->err = fabs(cnst_tri[n]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						cnst_tri[n]->hosted_dem.push_back(tmp_dem);
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break;
 					}
 				}
@@ -783,6 +927,11 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 						tmp_dem->host = cnst_tri[n];
 						tmp_dem->err = fabs(cnst_tri[n]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						cnst_tri[n]->hosted_dem.push_back(tmp_dem);
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break;
 					}
 				}
@@ -792,7 +941,7 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 			{
 				if (cnst_tri[n] != nullptr)
 				{
-					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_point);
+					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_error);
 				}
 			}
 
@@ -816,6 +965,11 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 						tmp_dem->host = cnst_tri[n];
 						tmp_dem->err = fabs(cnst_tri[n]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						cnst_tri[n]->hosted_dem.push_back(tmp_dem);
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break;
 					}
 				}
@@ -825,7 +979,7 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
 			{
 				if (cnst_tri[n] != nullptr)
 				{
-					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_point);
+					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_error);
 				}
 			}
 
@@ -917,15 +1071,39 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
  * @param[in]  err_records  If this pointer is not NULL, record maximal error values after each insertion of vertex.
  */
 void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_verts, std::vector<triangle*> &out_tris, 
-	double maxi_err, std::vector<vertex2dc> *outline_poly, std::vector<double> *err_records)
+	double maxi_err, std::vector<vertex2dc> *outline_poly, std::vector<double> *err_records, 
+	std::vector<region> *regions, std::vector<double> *dem_error)
 {
-	if (dem.size() < 3) return;
-	if (maxi_err <= 0.0) return;
+	if (dem.size() < 3)
+	{
+		throw std::runtime_error("The input DEM must have at least 3 points.");
+	}
+
+	if (maxi_err <= 0.0)
+	{
+		throw std::invalid_argument("Invalid maximal allowed error.");
+	}
 
 	if (!out_verts.empty()) out_verts.clear();
 	if (!out_tris.empty())  out_tris.clear();
 	if (err_records != nullptr && !err_records->empty()) err_records->clear();
 
+	if (dem_error != nullptr)
+	{
+		if (dem_error->size() != dem.size())
+		{
+			throw std::invalid_argument("Invalid size of the input DEM errors w.r.t. the DEM parameters.");
+		}
+
+		for (size_t i = 0; i < dem.size(); i++)
+		{
+			if (dem_error->at(i) <= 0.0)
+			{
+				throw std::runtime_error("Invalid input DEM error value.");		
+			}
+		}
+	}
+
 	// locate the surrounding box and initiate the staring two triangles
 	double xmin = dem[0].x, xmax = dem[0].x;
 	double ymin = dem[0].y, ymax = dem[0].y;
@@ -972,7 +1150,10 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 		out_tris.push_back(tmp_tri); tmp_tri = nullptr;
 	}
 
-	if (out_tris.size() != 2) return;
+	if (out_tris.size() != 2)
+	{
+		throw std::runtime_error("Fail to construct the initial triangles.");
+	}
 
 	out_tris[0]->set_neighbor(nullptr, nullptr, out_tris[1]);
 	out_tris[1]->set_neighbor(out_tris[0], nullptr, nullptr);
@@ -995,6 +1176,29 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 					tmp_dem->host = out_tris[t];
 					tmp_dem->err = fabs(out_tris[t]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 					out_tris[t]->hosted_dem.push_back(tmp_dem);
+					// Find region
+					if (regions != nullptr)
+					{
+						for (size_t r = 0; r < regions->size(); r++)
+						{
+							if (location_inside_polygon(tmp_dem->x, tmp_dem->y, &(regions->at(r).poly_vert)))
+							{
+								tmp_dem->allowed_err = regions->at(r).maxi_err;
+								break;
+							}
+						}
+					}
+
+					if (dem_error != nullptr)
+					{
+						tmp_dem->allowed_err = dem_error->at(d);
+					}
+
+					// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+					if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+					{
+						tmp_dem->err = 0.0;
+					}
 					break; // already found, no need to search more
 				}
 			}
@@ -1014,6 +1218,29 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 						tmp_dem->host = out_tris[t];
 						tmp_dem->err = fabs(out_tris[t]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						out_tris[t]->hosted_dem.push_back(tmp_dem);
+						// Find region
+						if (regions != nullptr)
+						{
+							for (size_t r = 0; r < regions->size(); r++)
+							{
+								if (location_inside_polygon(tmp_dem->x, tmp_dem->y, &(regions->at(r).poly_vert)))
+								{
+									tmp_dem->allowed_err = regions->at(r).maxi_err;
+									break;
+								}
+							}
+						}
+
+						if (dem_error != nullptr)
+						{
+							tmp_dem->allowed_err = dem_error->at(d);
+						}
+
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break; // already found, no need to search more
 					}
 				}
@@ -1025,7 +1252,7 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 	double maxi_err_tmp = -1.0;
 	for (int t = 0; t < out_tris.size(); ++t)
 	{
-		std::sort(out_tris[t]->hosted_dem.begin(), out_tris[t]->hosted_dem.end(), compare_dem_point);
+		std::sort(out_tris[t]->hosted_dem.begin(), out_tris[t]->hosted_dem.end(), compare_dem_error);
 		if (out_tris[t]->hosted_dem[0]->err > maxi_err_tmp)
 		{
 			maxi_err_tmp = out_tris[t]->hosted_dem[0]->err; 
@@ -1103,6 +1330,11 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 						tmp_dem->host = cnst_tri[n];
 						tmp_dem->err = fabs(cnst_tri[n]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						cnst_tri[n]->hosted_dem.push_back(tmp_dem);
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break;
 					}
 				}
@@ -1118,6 +1350,11 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 						tmp_dem->host = cnst_tri[n];
 						tmp_dem->err = fabs(cnst_tri[n]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						cnst_tri[n]->hosted_dem.push_back(tmp_dem);
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break;
 					}
 				}
@@ -1127,7 +1364,7 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 			{
 				if (cnst_tri[n] != nullptr)
 				{
-					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_point);
+					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_error);
 				}
 			}
 
@@ -1151,6 +1388,11 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 						tmp_dem->host = cnst_tri[n];
 						tmp_dem->err = fabs(cnst_tri[n]->interpolate(tmp_dem->x, tmp_dem->y) - tmp_dem->elev);
 						cnst_tri[n]->hosted_dem.push_back(tmp_dem);
+						// If the dem point belongs to a region, then set err to 0 if satisfy the region's threshold 
+						if (tmp_dem->allowed_err > 0.0 && tmp_dem->err <= tmp_dem->allowed_err)
+						{
+							tmp_dem->err = 0.0;
+						}
 						break;
 					}
 				}
@@ -1160,7 +1402,7 @@ void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_ver
 			{
 				if (cnst_tri[n] != nullptr)
 				{
-					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_point);
+					std::sort(cnst_tri[n]->hosted_dem.begin(), cnst_tri[n]->hosted_dem.end(), compare_dem_error);
 				}
 			}
 
diff --git a/src/lib/tin.h b/src/lib/tin.h
index f2cd7c1..1c2926d 100644
--- a/src/lib/tin.h
+++ b/src/lib/tin.h
@@ -14,16 +14,14 @@
 
 #ifndef _TIN_DELAUNAY_H
 #define _TIN_DELAUNAY_H
-#include "cmath"
 #include "vector"
-#include "algorithm"
 
 #define ZERO 1e-5
 
 // Start vertex definition
 struct vertex2dc
 {
-	unsigned int id; // index of the vertex
+	size_t id; // index of the vertex
 	double x, y; // position of the vertex
 	double elev; // elevation at the vertex
 
@@ -37,7 +35,7 @@ struct vertex2dc
 	 * @param inelev   initial elevation
 	 * @param inid     initial index
 	 */
-	vertex2dc(double inx, double iny, double inelev, unsigned int inid);
+	vertex2dc(double inx, double iny, double inelev, size_t inid);
 
 	/**
 	 * @brief Set a vertex2dc object
@@ -47,7 +45,7 @@ struct vertex2dc
 	 * @param inelev   initial elevation
 	 * @param inid     initial index
 	 */
-	void set(double inx, double iny, double inelev, unsigned int inid);
+	void set(double inx, double iny, double inelev, size_t inid);
 };
 
 /**
@@ -60,6 +58,16 @@ struct vertex2dc
  */
 bool operator ==(const vertex2dc &a, const vertex2dc &b);
 
+/**
+ * @brief Compare two vertexes to see if they are duplicated.
+ * 
+ * @param a       vertex a
+ * @param b       vertex b
+ * @return true   the two vertexes are at the same location and a same index
+ * @return false  the two vertexes are not at the same location  and a same index
+ */
+bool duplicated_vertex(const vertex2dc &a, const vertex2dc &b);
+
 /**
  * @brief Test if the three points are on the same line
  * 
@@ -84,6 +92,20 @@ bool is_collinear(vertex2dc *a_ptr, vertex2dc *b_ptr, vertex2dc *c_ptr);
 void circumcircle(vertex2dc *v0, vertex2dc *v1, vertex2dc *v2, double &cx, double &cy, double &cr);
 // End vertex definition
 
+// Start region definition
+struct region
+{
+	double maxi_err;
+	std::vector<vertex2dc> poly_vert;
+
+	region();
+
+	region(const std::vector<vertex2dc> &outline_vert, double err);
+
+	void set(const std::vector<vertex2dc> &outline_vert, double err);
+};
+// End region definition
+
 // Start DEM definition
 struct triangle;
 
@@ -92,6 +114,7 @@ struct dem_point
 	double x, y; // position of the DEM location
 	double elev; // elevation at the DEM location
 	double err; // error of the TIN with respect to the elevation
+	double allowed_err; // point specified threshold of maximal allowed error
 	triangle *host; // pointer of the triangle that the dem_point falls inside of
 
 	dem_point();
@@ -152,7 +175,7 @@ struct triangle
 	 * @param v1ptr   pointer of the vertex 1
 	 * @param v2ptr   pointer of the vertex 2
 	 */
-	void set(vertex2dc *v0ptr, vertex2dc *v1ptr, vertex2dc *v2ptr);
+	void set_vertex(vertex2dc *v0ptr, vertex2dc *v1ptr, vertex2dc *v2ptr);
 
 	/**
 	 * @brief Set neighbors of a triangle object
@@ -202,7 +225,8 @@ struct triangle
  */
 void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ymin, double ymax, 
 	double dx, double dy, std::vector<vertex2dc*> &out_verts, std::vector<triangle*> &out_tris, 
-	double maxi_err = 1e-0, std::vector<vertex2dc> *outline_poly = nullptr, std::vector<double> *err_records = nullptr);
+	double maxi_err = 1e-0, std::vector<vertex2dc> *outline_poly = nullptr, std::vector<double> *err_records = nullptr, 
+	std::vector<region> *regions = nullptr, std::vector<double> *dem_error = nullptr);
 
 /**
  * @brief      Generate the TIN from random DEM points
@@ -216,6 +240,7 @@ void grd2tin(const std::vector<double> &dem, double xmin, double xmax, double ym
  */
 void rnd2tin(const std::vector<dem_point> &dem, std::vector<vertex2dc*> &out_verts, 
 	std::vector<triangle*> &out_tris, double maxi_err = 1e-0, std::vector<vertex2dc> *outline_poly = nullptr, 
-	std::vector<double> *err_records = nullptr);
+	std::vector<double> *err_records = nullptr, std::vector<region> *regions = nullptr, 
+	std::vector<double> *dem_error = nullptr);
 
 #endif // _TIN_DELAUNAY_H
\ No newline at end of file
diff --git a/src/tool/grd2tin.cpp b/src/tool/grd2tin.cpp
index 3432b28..8ec24fd 100644
--- a/src/tool/grd2tin.cpp
+++ b/src/tool/grd2tin.cpp
@@ -5,6 +5,7 @@
 #include "fstream"
 #include "sstream"
 #include "string"
+#include "cmath"
 #include "getopt.h"
 
 void display_help(std::string exe_name)