diff --git a/src/render/CMakeLists.txt b/src/render/CMakeLists.txt
index 330356f..64b8e69 100644
--- a/src/render/CMakeLists.txt
+++ b/src/render/CMakeLists.txt
@@ -2,7 +2,8 @@ find_package(VTK REQUIRED COMPONENTS CommonCore CommonDataModel FiltersGeometry
 find_package(GDAL CONFIG REQUIRED)
 add_library(geopro_render STATIC
   Scene.cpp ColorLutBuilder.cpp CameraPreset.cpp VoxelFromScatters.cpp ContourBands.cpp actors/GridContourActor.cpp actors/VoxelActor.cpp actors/CurtainActor.cpp actors/MapLineActor.cpp actors/ScatterActor.cpp actors/AnomalyActor.cpp actors/ElectrodeActor.cpp actors/TerrainActor.cpp actors/AxesActor.cpp
-  interact/SlicePlaneMath.cpp interact/SliceTool.cpp interact/PickInteractorStyle.cpp interact/InteractionManager.cpp)
+  interact/SlicePlaneMath.cpp interact/SliceTool.cpp interact/PickInteractorStyle.cpp interact/InteractionManager.cpp
+  ground/TileMath.cpp)
 target_include_directories(geopro_render PUBLIC ${CMAKE_CURRENT_SOURCE_DIR})
 target_link_libraries(geopro_render PUBLIC geopro_core ${VTK_LIBRARIES} GDAL::GDAL)
 target_compile_features(geopro_render PUBLIC cxx_std_17)
diff --git a/src/render/ground/TileMath.cpp b/src/render/ground/TileMath.cpp
new file mode 100644
index 0000000..3b3b65d
--- /dev/null
+++ b/src/render/ground/TileMath.cpp
@@ -0,0 +1,35 @@
+#include "ground/TileMath.hpp"
+
+#include <cmath>
+
+namespace geopro::render {
+
+namespace {
+constexpr double kPi = 3.14159265358979323846;
+int clampInt(int v, int lo, int hi) { return v < lo ? lo : (v > hi ? hi : v); }
+}  // namespace
+
+TileXY lonLatToTile(double lonDeg, double latDeg, int z) {
+    if (z < 0) z = 0;
+    const double n = std::pow(2.0, z);
+    const double latR = latDeg * kPi / 180.0;
+    int x = static_cast<int>(std::floor((lonDeg + 180.0) / 360.0 * n));
+    int y = static_cast<int>(std::floor((1.0 - std::asinh(std::tan(latR)) / kPi) / 2.0 * n));
+    const int hi = static_cast<int>(n) - 1;
+    return TileXY{z, clampInt(x, 0, hi), clampInt(y, 0, hi)};
+}
+
+LonLatBox tileBounds(int z, int x, int y) {
+    if (z < 0) z = 0;
+    const double n = std::pow(2.0, z);
+    const double west = x / n * 360.0 - 180.0;
+    const double east = (x + 1) / n * 360.0 - 180.0;
+    auto latAt = [&](double yy) {
+        return std::atan(std::sinh(kPi * (1.0 - 2.0 * yy / n))) * 180.0 / kPi;
+    };
+    const double north = latAt(static_cast<double>(y));
+    const double south = latAt(static_cast<double>(y + 1));
+    return LonLatBox{west, south, east, north};
+}
+
+}  // namespace geopro::render
diff --git a/src/render/ground/TileMath.hpp b/src/render/ground/TileMath.hpp
new file mode 100644
index 0000000..aba1be8
--- /dev/null
+++ b/src/render/ground/TileMath.hpp
@@ -0,0 +1,22 @@
+#pragma once
+
+// Web Mercator(EPSG:3857) 瓦片坐标数学：天地图/XYZ 底图瓦片定位用（纯函数，无 VTK/Qt 依赖）。
+// 标准 slippy-map 公式：n=2^z；x=(lon+180)/360*n；y 用墨卡托纬度映射。
+namespace geopro::render {
+
+struct TileXY {
+    int z = 0, x = 0, y = 0;
+};
+
+// 瓦片地理边界（度）：west/east 经度，south/north 纬度。
+struct LonLatBox {
+    double west = 0, south = 0, east = 0, north = 0;
+};
+
+// 经纬度(度) → 指定 zoom 的瓦片行列（x/y 夹紧到 [0, 2^z-1]）。
+TileXY lonLatToTile(double lonDeg, double latDeg, int z);
+
+// 瓦片 (z,x,y) → 其覆盖的地理边界（度）。
+LonLatBox tileBounds(int z, int x, int y);
+
+}  // namespace geopro::render
diff --git a/tests/CMakeLists.txt b/tests/CMakeLists.txt
index 4bc8839..7c5661d 100644
--- a/tests/CMakeLists.txt
+++ b/tests/CMakeLists.txt
@@ -100,6 +100,8 @@ target_sources(geopro_tests PRIVATE render/test_terrain.cpp)
 target_sources(geopro_tests PRIVATE render/test_camera_preset.cpp)
 # AxesActor(P2)：buildAxes(bounds+unit/mode→vtkCubeAxesActor) 单位换算(英尺/经纬度)/不显示返回空。
 target_sources(geopro_tests PRIVATE render/test_axes.cpp)
+# TileMath(P5)：天地图底图 Web Mercator 瓦片坐标数学(经纬↔z/x/y、瓦片地理边界)——纯函数。
+target_sources(geopro_tests PRIVATE render/test_tile_math.cpp)
 # SlicePlaneMath(P3)：切面法向/滚轮平移+夹限/双击正视相机(含竖直兜底)/滚轮步长/最近切片——纯几何。
 target_sources(geopro_tests PRIVATE render/test_slice_plane_math.cpp)
 target_link_libraries(geopro_tests PRIVATE geopro_render ${VTK_LIBRARIES})
diff --git a/tests/render/test_tile_math.cpp b/tests/render/test_tile_math.cpp
new file mode 100644
index 0000000..5575f5e
--- /dev/null
+++ b/tests/render/test_tile_math.cpp
@@ -0,0 +1,48 @@
+#include <gtest/gtest.h>
+
+#include <cmath>
+
+#include "ground/TileMath.hpp"
+
+using geopro::render::lonLatToTile;
+using geopro::render::tileBounds;
+
+// z=1 把世界分 2x2：原点(0°,0°)在东/南象限交界 → 标准 slippy 取 (1,1)。
+TEST(TileMath, OriginZoom1) {
+    auto t = lonLatToTile(0.0, 0.0, 1);
+    EXPECT_EQ(t.z, 1);
+    EXPECT_EQ(t.x, 1);
+    EXPECT_EQ(t.y, 1);
+}
+
+// z=1 西北瓦片 (0,0) 覆盖西半球北部：west=-180, east=0, north≈85.0511(墨卡托上限), south=0。
+TEST(TileMath, BoundsZoom1NW) {
+    auto b = tileBounds(1, 0, 0);
+    EXPECT_NEAR(b.west, -180.0, 1e-6);
+    EXPECT_NEAR(b.east, 0.0, 1e-6);
+    EXPECT_NEAR(b.north, 85.0511287798, 1e-4);
+    EXPECT_NEAR(b.south, 0.0, 1e-6);
+}
+
+// 往返一致：任一经纬点所属瓦片的边界必须包含该点（经度严格、纬度含墨卡托方向）。
+TEST(TileMath, RoundTripContains) {
+    const double lon = 116.391, lat = 39.907;  // 北京附近
+    const int z = 12;
+    auto t = lonLatToTile(lon, lat, z);
+    EXPECT_EQ(t.z, z);
+    auto b = tileBounds(t.z, t.x, t.y);
+    EXPECT_GE(lon, b.west);
+    EXPECT_LE(lon, b.east);
+    EXPECT_LE(lat, b.north);  // north 是瓦片上边界(纬度大)
+    EXPECT_GE(lat, b.south);
+}
+
+// 夹紧：超界经纬不应产生越界瓦片索引。
+TEST(TileMath, ClampInRange) {
+    auto t = lonLatToTile(500.0, 95.0, 3);  // 非法输入
+    const int hi = (1 << 3) - 1;
+    EXPECT_GE(t.x, 0);
+    EXPECT_LE(t.x, hi);
+    EXPECT_GE(t.y, 0);
+    EXPECT_LE(t.y, hi);
+}