OpenGL技术问题：如何渲染纹理区域外的内容

我正在实现一款可为纹理添加圆角与边框的片段着色器，用于窗口管理器实现带边框的圆角窗口。当前片段着色器代码如下：

const GLchar surface_fragment_src_rgba[] = 
 "precision mediump float;\n" 
 "varying vec2 v_texcoord;\n" 
 "uniform sampler2D tex;\n" 
 "\n" 
 "uniform vec2 half_size;\n" 
 "uniform vec2 position;\n" 
 "uniform float alpha;\n" 
 "uniform float radius;\n" 
 "uniform float half_border_thickness;\n" 
 "uniform vec4 border_color;\n" 
 "\n" 
 "float RectSDF(vec2 p, vec2 b, float r) {\n" 
 "   vec2 d = abs(p) - b + vec2(r);\n" 
 "        return min(max(d.x, d.y), 0.0) + length(max(d, 0.0)) - r;\n" 
 "}\n" 
 "\n" 
 "void main() {\n" 
 "        vec2 center = gl_FragCoord.xy - position - half_size;\n" 
 "        float dist = RectSDF(center, half_size + half_border_thickness, radius - half_border_thickness);\n" 
 "\n" 
 "        vec4 tex_color  = texture2D(tex, v_texcoord) * alpha;\n" 
 "   vec4 from_color = border_color;\n" 
 "   vec4 to_color   = vec4(0.0);\n" 
 "        if (half_border_thickness > 0.0) {\n" 
 "                if (dist < 0.0) {\n" 
 "                        to_color = tex_color;\n" 
 "                }\n" 
 "                dist = abs(dist) - half_border_thickness;\n" 
 "        } else {\n" 
 "                from_color = tex_color;\n" 
 "        }\n" 
 "\n" 
 "        float blend_amount = smoothstep(-1.0, 1.0, dist);\n" 
 "        gl_FragColor = mix(from_color, to_color, blend_amount);\n" 
 "}\n";;

我在Shadertoy上制作了验证原型，确认着色器逻辑本身符合预期，但实际渲染窗口时，纹理原始边界外的区域无法被正确渲染（表现为边框在纹理边缘处断裂）。求问如何实现完整的窗口区域加边框渲染？

另外，如果当前的实现方式存在优化空间，也欢迎指出。最初我用单独的着色器渲染边框，但为了实现抗锯齿圆角的边框混合效果，我认为必须获取纹理颜色信息，所以改成了当前的实现方式。

内容的提问来源于stack exchange，提问作者willmckinnon